
Habitat loss and fragmentation is the largest contributing factor to species extinction and declining biodiversity. Landscapes are becoming highly spatially heterogeneous with varying degrees of human modification. Much theoretical study of habitat fragmentation has historically focused on a simple theoretical landscape with patches of habitat surrounded by a spatially homogeneous hostile matrix. However, terrestrial habitat patches are often surrounded by complex mosaics of many different land cover types, which are rarely ecologically neutral or completely inhospitable environments. We employ an extension of a reaction diffusion model to explore effects of heterogeneity in the matrix immediately surrounding a patch in a one-dimensional theoretical landscape. Exact dynamics of a population exhibiting logistic growth, an unbiased random walk in the patch and matrix, habitat preference at the patch/matrix interface, and two functionally different matrix types for the one-dimensional landscape is obtained. These results show existence of a minimum patch size (MPS), below which population persistence is not possible. This MPS can be estimated via empirically derived estimates of patch intrinsic growth rate and diffusion rate, habitat preference, and matrix death and diffusion rates. We conclude that local matrix heterogeneity can greatly change model predictions, and argue that conservation strategies should not only consider patch size, configuration, and quality, but also quality and spatial structure of the surrounding matrix.
Citation: Nalin Fonseka, Jerome Goddard Ⅱ, Alketa Henderson, Dustin Nichols, Ratnasingham Shivaji. Modeling effects of matrix heterogeneity on population persistence at the patch-level[J]. Mathematical Biosciences and Engineering, 2022, 19(12): 13675-13709. doi: 10.3934/mbe.2022638
[1] | Nazanin Zaker, Christina A. Cobbold, Frithjof Lutscher . The effect of landscape fragmentation on Turing-pattern formation. Mathematical Biosciences and Engineering, 2022, 19(3): 2506-2537. doi: 10.3934/mbe.2022116 |
[2] | Robert Stephen Cantrell, Chris Cosner, William F. Fagan . The implications of model formulation when transitioning from spatial to landscape ecology. Mathematical Biosciences and Engineering, 2012, 9(1): 27-60. doi: 10.3934/mbe.2012.9.27 |
[3] | James T. Cronin, Jerome Goddard II, Amila Muthunayake, Ratnasingham Shivaji . Modeling the effects of trait-mediated dispersal on coexistence of mutualists. Mathematical Biosciences and Engineering, 2020, 17(6): 7838-7861. doi: 10.3934/mbe.2020399 |
[4] | Yongli Cai, Yun Kang, Weiming Wang . Global stability of the steady states of an epidemic model incorporating intervention strategies. Mathematical Biosciences and Engineering, 2017, 14(5&6): 1071-1089. doi: 10.3934/mbe.2017056 |
[5] | Liqiong Pu, Zhigui Lin . A diffusive SIS epidemic model in a heterogeneous and periodically evolvingenvironment. Mathematical Biosciences and Engineering, 2019, 16(4): 3094-3110. doi: 10.3934/mbe.2019153 |
[6] | James T. Cronin, Nalin Fonseka, Jerome Goddard II, Jackson Leonard, Ratnasingham Shivaji . Modeling the effects of density dependent emigration, weak Allee effects, and matrix hostility on patch-level population persistence. Mathematical Biosciences and Engineering, 2020, 17(2): 1718-1742. doi: 10.3934/mbe.2020090 |
[7] | Maryam Basiri, Frithjof Lutscher, Abbas Moameni . Traveling waves in a free boundary problem for the spread of ecosystem engineers. Mathematical Biosciences and Engineering, 2025, 22(1): 152-184. doi: 10.3934/mbe.2025008 |
[8] | Guilherme M Lopes, José F Fontanari . Influence of technological progress and renewability on the sustainability of ecosystem engineers populations. Mathematical Biosciences and Engineering, 2019, 16(5): 3450-3464. doi: 10.3934/mbe.2019173 |
[9] | Thomas G. Hallam, Qingping Deng . Simulation of structured populations in chemically stressed environments. Mathematical Biosciences and Engineering, 2006, 3(1): 51-65. doi: 10.3934/mbe.2006.3.51 |
[10] | Md. Mashih Ibn Yasin Adan, Md. Kamrujjaman, Md. Mamun Molla, Muhammad Mohebujjaman, Clarisa Buenrostro . Interplay of harvesting and the growth rate for spatially diversified populations and the testing of a decoupled scheme. Mathematical Biosciences and Engineering, 2023, 20(4): 6374-6399. doi: 10.3934/mbe.2023276 |
Habitat loss and fragmentation is the largest contributing factor to species extinction and declining biodiversity. Landscapes are becoming highly spatially heterogeneous with varying degrees of human modification. Much theoretical study of habitat fragmentation has historically focused on a simple theoretical landscape with patches of habitat surrounded by a spatially homogeneous hostile matrix. However, terrestrial habitat patches are often surrounded by complex mosaics of many different land cover types, which are rarely ecologically neutral or completely inhospitable environments. We employ an extension of a reaction diffusion model to explore effects of heterogeneity in the matrix immediately surrounding a patch in a one-dimensional theoretical landscape. Exact dynamics of a population exhibiting logistic growth, an unbiased random walk in the patch and matrix, habitat preference at the patch/matrix interface, and two functionally different matrix types for the one-dimensional landscape is obtained. These results show existence of a minimum patch size (MPS), below which population persistence is not possible. This MPS can be estimated via empirically derived estimates of patch intrinsic growth rate and diffusion rate, habitat preference, and matrix death and diffusion rates. We conclude that local matrix heterogeneity can greatly change model predictions, and argue that conservation strategies should not only consider patch size, configuration, and quality, but also quality and spatial structure of the surrounding matrix.
The largest contributing factor to species extinction and declining biodiversity is habitat loss and fragmentation e.g., [1,2,3,4,5]. Across terrestrial ecosystems worldwide, landscapes are becoming highly spatially heterogeneous with varying degrees of human modification [6]. Many studies show that fragmentation is often associated with significant changes in density and that individual species can be differentially affected by fragmentation [7,8,9,10,11]. In fact, understanding effects of fragmentation has become a central focus of how to guide conservation efforts [12].
Much habitat fragmentation research has historically been based upon the theories of island biogeography and metapopulation dynamics. Both theories assume a simple binary representation of landscapes consisting either of habitat patches or inhospitable, homogeneous matrix, which is viewed as unimportant [13]. This dichotomous view of a landscape has been the guide for most fragmentation research in the past decades [13]. Researchers have placed much emphasis on patch-level attributes such as patch-size and largely ignored heterogeneity of the landscape context (i.e., matrix) [12,13,14,15]. As such, their ability to make accurate predictions of real-world patterns of patch occurrence, population persistence, and dispersal has been diminished [6]. The relative importance of landscape heterogeneity and matrix composition compared to focal habitat configuration and composition remains an unanswered question [16].
Notwithstanding, terrestrial habitat patches are often surrounded by complex mosaics of many different land cover types, which are rarely ecologically neutral or completely inhospitable environments [16]. Edge effects are also predicted to play a major role in shaping population dynamics [17]. Even with an early mention in the literature, incorporation of effects of a heterogeneous matrix into habitat fragmentation study has only recently been made [12]. Even though this has been discussed by landscape ecologists, only a few authors have attempted to develop models which simulate effects of a heterogeneous matrix on population persistence and movement patterns in hypothetical landscapes [12]. These models often employ arbitrarily chosen values for matrix characteristics and lack a mechanistic underpinning.
Connecting the wealth of empirical information available about individual movement and mortality in response to matrix composition to predictions about patch-level persistence is a formidable task [18]. The sheer complexity of realistic landscapes has hampered efforts to develop modeling frameworks capable of capturing all important spatial features [12]. However, adaptation of the reaction diffusion framework has seen some recent success in incorporating certain aspects of matrix heterogeneity into the model in order to study population persistence at the patch-level. Recently, the authors have developed a modeling framework built upon reaction diffusion models to study effects of variable matrix hostility and habitat loss on population persistence at the patch-level. The framework extends pioneering work done by authors who first attempted to incorporate matrix hostility and edge effects into reaction diffusion models for organisms dwelling in a one-dimensional patch-matrix system [18,19,20] to two- and three-dimensional landscapes [21]. The framework provides a mechanistic connection of individual-level assumptions regarding dispersal in the patch and matrix, growth processes in the patch, hostility in the matrix, and habitat preference at the patch/matrix interface to patch-level predictions of population persistence [21]. Subsequent studies employing this framework have uncovered important consequences of matrix hostility, edge effects, and habitat preference on population persistence [22,23,24,25,26].
This framework has allowed a more realistic study of landscapes in which each patch can be modeled with different assumptions regarding movement, hostility, and habitat preference in the local matrix surrounding the patch. Even though landscape heterogeneity is incorporated into the framework, there is an implicit assumption that the matrix immediately surrounding a patch is spatially homogeneous. We define this type of spatial homogeneity within a heterogeneous landscape as locally homogeneous matrix. The present work is an attempt to extend the framework to the case where spatial heterogeneity is present even within the matrix immediately surrounding a patch, which we define as locally heterogeneous matrix. In particular, we envision a patch surrounded by a locally heterogeneous matrix with two functionally different matrix types (as in [27], functionally different refers to differences noticeable to the organism of interest which may or may not be noticeable to humans). Figure 1 illustrates this scenario with a forested patch (Ω) surrounded by active farmland on the left (ΩM1) and inactive farmland on the right (ΩM2).
We initiate this study by focusing on investigation of population persistence in the presence of spatial heterogeneity in the matrix for a one-dimensional analog of the landscape illustrated in Figure 1 (see Figure 2).
Here, we describe the modeling framework for our one-dimensional patch-matrix system. The model is built upon the reaction diffusion framework which has seen tremendous success in studying spatially structured systems (see [28,29,30,31,32,33,34] and references therein for a detailed history of the framework). We make the assumption that a single species is dwelling in a focal patch ˜Ω=(0,ℓ) with patch size ℓ>0 that is surrounded by a locally heterogeneous matrix (˜ΩM) with two functionally different components denoted by ˜ΩM1=(−∞,0) and ˜ΩM2=(ℓ,∞). We further assume that organisms disperse with an unbiased random walk with diffusion rate D0i>0 and experience exponential decay at fixed rate S0i>0 for each matrix component ˜ΩMi, respectively, i=1,2. Denote the boundary of ˜Ω by ∂˜Ω={0,ℓ}. The variables t and x represent time and spatial location within the patch. Inside the patch, organisms are assumed to follow logistic growth and an unbiased random walk, while at the patch/matrix interface a discontinuity between the density in the patch and matrix is allowed (via a biased random walk). However, continuity in the flux is assumed, as has been shown to arise naturally from a landscape-level random walk derivation [18,20,35]. In this case, organisms recognize the patch/matrix interface and potentially modify their random walk movement probability (i.e., probability of an organism moving at a given time step in the random walk process), random walk step length (i.e., distance that an organism moves during a given time step), and/or probability of remaining in the patch (say αi). Here, we equate dispersal from the patch to organisms reaching the patch/matrix interface, leaving the patch with probability 1−αi (taken to be constant), and entering the matrix, where they still have the opportunity to re-enter the patch at the interface. A straightforward modification of the derivation given in [21], gives the following reaction diffusion model:
{ut=Duxx+ru(1−uK);t>0,x∈(0,ℓ)u(0,x)=u0(x);x∈(0,ℓ)−Dα1ux(t,0)+S∗1[1−α1]u(t,0)=0;t>0Dα2ux(t,ℓ)+S∗2[1−α2]u(t,ℓ)=0;t>0 | (1.1) |
which will exactly model the described study system in the sense that steady states of (1.1) and their stability properties will be exactly the same as those of the study system (see [21] and references therein). Here, D>0 represents patch diffusion rate, r>0 patch intrinsic growth rate, K>0 patch carrying capacity, u0(x) initial population density distribution in the patch, and αi the probability of an individual staying in the patch upon reaching the boundary interfacing with ˜ΩMi. Note that the parameter S∗i=√S0iD0iκi≥0 represents the effective matrix hostility towards an organism in ˜ΩMi, has units of length by time, and can assume different forms depending upon the patch/matrix interface assumptions as encoded in κi. Table 1 shows different cases for κi corresponding to different patch/matrix interface assumptions as derived in [21]. Notice that when αi≡0, the boundary interfacing with ˜ΩMi is absorbing, i.e., all individuals that reach the boundary will emigrate into ˜ΩMi, while, αi≡1, implies the boundary interfacing with ˜ΩMi is reflecting, i.e., emigration rate into matrix ˜ΩMi is zero. See Table 2 for a summary of model parameters.
Scenario Name | Scenario Description | κi | References |
Continuous Density | Organisms move between the patch and matrix with equal probability. Step sizes and movement probabilities are equal in the patch and matrix. | 1 | [36] |
Type Ⅰ Discontinuous Density (DD) | Organisms modify their movement behavior at the patch/matrix interface and would have a probability α of remaining in ˜Ω which may be different from 50%. Step sizes differ between the patch and matrix, whereas movement probabilities are equal. | √D0iD | [18,20] |
Type Ⅱ Discontinuous Density (DD) | Organisms modify their movement behavior at the patch/matrix interface and would have a probability α of remaining in ˜Ω which may be different from 50%. Step sizes are equal between the patch and matrix but movement probabilities are different. | D0iD | [18,20] |
Type Ⅲ Discontinuous Density (DD) | Organisms remain in ˜Ω with probability α which may be different from 50%. Movement probabilities and step sizes are the same between the patch and matrix. | 1 | [37,38] |
Parameter | Definition | Units |
ℓ | Length of patch ˜Ω | length |
r | Patch intrinsic growth rate | 1/time |
D | Patch diffusion rate | length2/time |
K | Patch carrying capacity | density |
S0i | Death rate in matrix component ˜ΩMi | 1/time |
D0i | Diffusion rate in matrix component ˜ΩMi | length2/time |
κi | Patch/matrix interface parameter for ˜ΩMi | unitless |
Introducing a standard scaling [21],
˜x=xℓ, ˜u=uK, ˜t=rt, | (1.2) |
and dropping the tilde, the patch becomes Ω=(0,1), left matrix becomes ΩM1, right matrix becomes ΩM2, and (1.1) becomes
{ut=1λuxx+u(1−u);t>0,x∈(0,1)u(0,x)=u0(x);x∈(0,1)−ux(t,0)+√λγ1u(t,0)=0;t>0ux(t,1)+√λγ2u(t,1)=0;t>0 | (1.3) |
with steady state equation
{−u″=λu(1−u);(0,1)−u′(0)+√λγ1u(0)=0u′(1)+√λγ2u(1)=0 | (1.4) |
where λ>0 is a composite parameter which is proportional to patch size squared and γi is a composite parameter proportional to effective matrix hostility in matrix component ΩMi. Table 3 gives a summary of these composite parameters and their definitions in the different scenarios outlined in Table 1, i.e., Continuous Density (CTS) and Type Ⅰ - Ⅲ Discontinuous Density (DD).
Composite Parameter | Expression | Range |
λ | rℓ2D | 0<λ<∞ |
CTS, Type Ⅰ DD, & Type Ⅲ DD: γi | √S0ir1−αiαi | 0≤γi<∞ |
Type Ⅱ DD: γi | √S0iDrD0i1−αiαi | 0≤γi<∞ |
In the present work, we consider the following two cases:
● Case 1: Effective matrix hostility in ΩM2 is fixed and finite, while allowed to vary in ΩM1.
● Case 2: Matrix component ΩM1 is immediately lethal to organisms, while effective matrix hostility is allowed to vary in ΩM2.
Notice that in Case 2, when γ1→∞ we have that (1.4) becomes:
{−u″=λu(1−u);(0,1)u(0)=0u′(1)+√λγ2u(1)=0. | (1.5) |
It is easy to see that similar results to Cases 1 & 2 hold when changing the matrix component with fixed hostility. We analyze the structure and stability properties of positive solutions for (1.4) in Case 1 and (1.5) in Case 2, providing a complete description of the dynamics of the system.
Here, we recall population dynamics of a single species (with density denoted as u) dwelling in a patch (Ω) located in heterogeneous landscape but with a locally homogeneous matrix (ΩM) as previously studied in [21,39]. Namely, the authors studied the structure and stability properties of positive steady state solutions of (1.1) in the case when the combined matrix hostility is the same, i.e., γ1=γ2=γ via:
{−u″=λu(1−u);(0,1)−u′(0)+√λγu(0)=0u′(1)+√λγu(1)=0. | (1.6) |
The authors gave a complete description of the dynamics of (1.1) in this case as summarized in Theorem 1.1.
Theorem 1.1 (see [22,39]). Let γ>0 be fixed. Then the following hold:
(1) if λ≤E1(γ) then (1.6) has no positive solution and the trivial solution is globally asymptotically stable;
(2) if λ>E1(γ) then (1.6) has a unique globally asymptotically stable positive solution uλ such that ‖uλ‖∞→1− as λ→∞, ‖uλ‖∞→0 as λ→E1(γ)+, and uλ is symmetric about the center of the patch (x=12).
These results also established an exact bifurcation diagram of positive solutions for (1.6) as illustrated in Figure 3.
Here, given a γ>0, we denote E1(γ)>0 as the principal eigenvalue of the problem:
{−ϕ″=Eϕ;(0,1)−ϕ′(0)+√Eγϕ(0)=0ϕ′(1)+√Eγϕ(1)=0 | (1.7) |
with corresponding eigenfunction ϕ chosen such that ϕ(x)>0; ¯Ω. Note that existence of this eigenvalue was discussed in [39]. The bifurcation diagram of positive solutions of (1.6) illustrated in Figure 3 shows existence of a minimum patch size given by:
ℓ∗(γ)=√E1(γ)Dr. | (1.8) |
In Section 2, we present some mathematical preliminaries, followed by our main results and biological conclusions in Section 3. We provide proofs of our main results in Section 4. Finally, we give closing remarks in Section 5 and provide proofs of some technical details of our arguments in the Appendix.
In this section, we introduce definitions of a sub- and supersolution of (1.4) and then an existence result. We also state several important lemmas that we will use to establish our results.
By a subsolution of (1.4) we mean ψ∈C2((0,1))∩C1([0,1]) that satisfies
{−ψ″≤λψ(1−ψ);(0,1)−ψ′(0)+√λγ1ψ(0)≤0ψ′(1)+√λγ2ψ(1)≤0. |
By a supersolution of (1.4) we mean Z∈C2((0,1))∩C1([0,1]) that satisfies
{−Z″≥λZ(1−Z);(0,1)−Z′(0)+√λγ1Z(0)≥0Z′(1)+√λγ2Z(1)≥0. |
By a strict subsolution of (1.4) we mean a subsolution which is not a solution. By a strict supersolution of (1.4) we mean a supersolution which is not a solution. The definitions of sub- and supersolution of (1.5) are analogous.
We now state the following well-known result [40]:
Lemma 2.1. Let ψ and Z be a sub- and supersolution of (1.4), respectively, such that ψ≤Z; ¯Ω. Then (1.4) has a solution u∈C2((0,1))∩C1([0,1]) such that u∈[ψ,Z]; ¯Ω.
Remark 2.1. An analogous result holds for (1.5).
Next, we state several lemmas that we will use to construct sub-super solutions. First, denote E1(γ1,γ2) as the principal eigenvalue of:
{−ϕ″=Eϕ;(0,1)−ϕ′(0)+√Eγ1ϕ(0)=0ϕ′(1)+√Eγ2ϕ(1)=0. | (2.1) |
Lemma 2.2. For γ1,γ2, and λ>0, let σλ(γ1,γ2) be the principal eigenvalue of the problem:
{−ϕ″=(λ+σ)ϕ;(0,1)−ϕ′(0)+√λγ1ϕ(0)=0ϕ′(1)+√λγ2ϕ(1)=0. | (2.2) |
Then σλ(γ1,γ2)<0 for λ>E1(γ1,γ2) and λ≈E1(γ1,γ2), and σλ(γ1,γ2)>0 for λ<E1(γ1,γ2) and λ≈E1(γ1,γ2). Furthermore, σλ(γ1,γ2)→0 as λ→E1(γ1,γ2). (See appendix for details).
Lemma 2.3. The principal eigencurve, β=β(μ), of the eigenvalue problem:
{−ϕ″=βϕ;(0,1)ϕ(0)=0ϕ′(1)=−μϕ(1) | (2.3) |
with μ>0, is increasing, concave, and limμ→∞β(μ)=ED1. Here, ED1 is the principal eigenvalue of
{−ϕ″=Eϕ;(0,1)ϕ(0)=0ϕ(1)=0. | (2.4) |
(See appendix for details)
Lemma 2.4. For λ>0, let ˜σλ(γ2) be the principal eigenvalue of the problem:
{−ϕ″=(λ+σ)ϕ;(0,1)ϕ(0)=0ϕ′(1)+√λγ2ϕ(1)=0, | (2.5) |
then ˜σλ(γ2)<0 for λ>~E1(γ2) and ˜σλ(γ2)>0 for λ<~E1(γ2). Further, ˜σλ(γ2)→0 as λ→~E1(γ2). Here, ~E1(γ2)(=limγ1→∞E1(γ1,γ2)) is the principal eigenvalue of:
{−ϕ″=Eϕ;(0,1)ϕ(0)=0ϕ′(1)+√Eγ2ϕ(1)=0. | (2.6) |
(See appendix for details).
In this section, we present analytical and numerical results for both of the aforementioned cases.
We begin with some analytical results for (1.4). The first result provides a means of computing a bifurcation diagram of positive solutions for (1.4). This is an extension of the quadrature method introduced for Dirichlet problems in [41], and for the case of linear boundary conditions with γ1=γ2 in [39].
Theorem 3.1. Let λ>0 and ρ∈(0,1). Then (1.4) has a positive solution u with ‖u‖∞=ρ if and only if there exist q1,q2∈(0,ρ) such that u(0)=q1, u(1)=q2, and λ,ρ,q1,q2 satisfy:
λ=12{∫ρq1dz√F(ρ)−F(z)+∫ρq2dz√F(ρ)−F(z)}2(=λ(ρ)say) | (3.1) |
and
{2[F(ρ)−F(q1)]=γ21q212[F(ρ)−F(q2)]=γ22q22 | (3.2) |
where F(z)=∫z0s(1−s)ds. Furthermore, (3.2) uniquely determines q1(=q1(ρ)), q2(=q2(ρ)) and λ(=λ(ρ)) defines a continuous function of ρ on (0,1).
Remark 3.1. We note that for λ>0 if u is a positive solution of (1.4) with ||u||∞=ρ∈(0,1), then there exists a unique x∗∈(0,1) such that u is symmetric about the point x∗, increasing on (0,x∗), decreasing on (x∗,1), u′(x∗)=0, and ||u||∞=u(x∗) where
x∗=∫ρq1dz√F(ρ)−F(z)∫ρq1dz√F(ρ)−F(z)+∫ρq2dz√F(ρ)−F(z). |
and q1=u(0), q2=u(1) are uniquely determined by (3.2) (see Figure 4).
The next analytical result gives a complete description of the dynamics of (1.3) in this case.
Theorem 3.2. Let γ2>0 be fixed. Then for all γ1>0 the following hold:
1) if λ≤E1(γ1,γ2) then (1.4) has no positive solution and the trivial solution is globally asymptotically stable;
2) if λ>E1(γ1,γ2) then (1.4) has a unique globally asymptotically stable positive solution uλ such that ‖uλ‖∞→1− as λ→∞ and ‖uλ‖∞→0 as λ→E1(γ1,γ2)+. In fact, uλ→1− uniformly on compact subsets of (0,1) as λ→∞.
Recall, E1(γ1,γ2) is the principal eigenvalue of:
{−ϕ″=Eϕ;(0,1)−ϕ′(0)+√Eγ1ϕ(0)=0ϕ′(1)+√Eγ2ϕ(1)=0. | (3.3) |
Figure 5 illustrates Theorem 3.2.
Remark 3.2. We show in the appendix that
E1(γ1,γ2)={[tan−1(γ1+γ21−γ1γ2)]2forγ1γ2<1[π+tan−1(γ1+γ21−γ1γ2)]2forγ1γ2>1π24forγ1γ2=1. | (3.4) |
Note that for a fixed γ2>0,
1) E1(γ1,γ2)∈([tan−1(γ2)]2,[π+tan−1(−1γ2)]2) for γ1>0;
2) E1(γ1,γ2)→[tan−1(γ2)]2 as γ1→0 and E1(γ1,γ2)→[π+tan−1(−1γ2)]2 as γ1→∞;
3) E1(γ1,γ2)→π24 as γ1→1γ2 and E1(1γ2,γ2)=π24;
4) E1(γ1,γ2) is continuous and increasing in γ1.
The next result establishes that positive solutions of (1.4) have an ordering with respect to γ1 for a fixed γ2 (see Figure 6).
Theorem 3.3. Let γ2>0 be fixed. Then for γ1>γ∗1>0, if wλ:=uλ(γ1,γ2) and vλ:=uλ(γ∗1,γ2) are positive solutions of (1.4), then wλ<vλ; [0,1].
The final analytical result for this case shows that if γ1≠γ2 then u(0)=q1≠q2=u(1). In other words, local matrix heterogeneity can cause density profiles to become asymmetric. Theorem 3.4 is illustrated in Figure 7.
Theorem 3.4. Let γ2>0 be fixed. If γ1>γ2 (γ1<γ2) and uλ is a positive solution of (1.4) with ‖uλ‖∞=ρ, uλ(0)=q1, and uλ(1)=q2, then q1<q2 (q1>q2). Further, if γ1=γ2, then q1=q2.
Remark 3.3. We note that q1<q2 implies that x∗∈(12,1), q1>q2 implies that x∗∈(0,12), and q1=q2 implies that x∗=12.
Next, we present numerically generated bifurcation diagrams of positive solutions for (1.4) using the following procedure:
1) Fix γ1,γ2>0 and define ρi=in+1;i=1,...,n, where n≥1 is the desired number of interpolation points.
2) Using fzero in MATLAB (The Mathworks, Inc. version: R2022a), numerically find the roots of (3.2), i.e., qi1=q1(ρi) and qi2=q2(ρi), for a given ρi.
3) The values of ρi,qi1,qi2 are then substituted into (3.1) and the corresponding λi-value is numerically computed using integral.
4) Repeating (2)–(3) for i=1,2,...,n, we obtain (λi,ρi) points, generating a bifurcation curve of λ vs. ρ=‖u‖∞ for positive solutions of (1.4).
In terms of computational complexity, a single bifurcation diagram was able to be computed in about one second using a standard laptop.
Employing Theorem 3.2, the model predicts a minimum patch size (MPS) below which organisms cannot colonize the patch. Using the definition of λ from Table 3, we can give an explicit form for the predicted MPS,
ℓ∗(γ1,γ2)=√E1(γ1,γ2)Dr. | (3.5) |
Remark 3.2 now gives limiting behavior of MPS with respect to γ1 for fixed γ2>0 as presented in Remark 3.4.
Remark 3.4. Let γ2>0 be fixed. Then we have:
1) limγ1→0+ℓ∗(γ1,γ2)=tan−1(γ2)√Dr(=ℓ∗(0,γ2)say);
2) limγ1→∞ℓ∗(γ1,γ2)=(π+tan−1(−1γ2))√Dr(=ℓ∗(∞,γ2)say);
3) ℓ∗(∞,γ2)−ℓ∗(0,γ2)=π2√Dr.
From Remark 3.2, it immediately follows that ℓ∗(γ1,γ2)∈[ℓ∗(0,γ2),ℓ∗(∞,γ2)] and the length of this interval is a constant independent of the value of γ2>0.
Figure 8 shows an evolution of bifurcation curves as γ1 varies with γ2=1 and r=D chosen for convenience of presentation. In Figure 8a, we consider γ1 going to zero from the right and observe that MPS approaches ℓ∗(0,1)=π4 from the right, shifting the bifurcation curves to the left. In Figure 8b, we consider γ1 approaching ∞ and observe MPS approaching ℓ∗(∞,1)=3π4 from the left as γ1→∞, this time shifting the bifurcation curves to the right. Further, we observe that MPS ranges over (π4,3π4), increases in γ1, and causes bifurcation curves to move to the right as γ1 increases.
In Figure 9, we provide the evolution of MPS (ℓ∗(γ1,γ2)) as γ1, γ2 both vary with r=D chosen for convenience of presentation. Here, we observe that for any fixed γ2>0 (γ1>0), ℓ∗(γ1,γ2) is an increasing function of γ1 (γ2). Furthermore, we observe that MPS approaches 0+ as γ1,γ2→0 and approaches √ED1=π as γ1,γ2→∞, where ED1 is the principal eigenvalue of (2.4). Recall that these facts were proved analytically and stated in Remark 3.2.
We begin by presenting some analytical results for (1.4) in the case when γ1→∞, namely (1.5).
The first result is a modification of the time map analysis presented in Theorem 3.1 for this case.
Theorem 3.5. Let λ>0 and ρ∈(0,1). Then (1.5) has a positive solution u such that ‖u‖∞=ρ if and only if there exists q∈(0,ρ) such that u(1)=q and λ,ρ,q satisfy:
λ=12{∫ρ0dz√F(ρ)−F(z)+∫ρqdz√F(ρ)−F(z)}2(=λ(ρ)say) | (3.6) |
and
2[F(ρ)−F(q)]=γ22q2. | (3.7) |
where F(z)=∫z0s(1−s)ds. Furthermore, (3.7) uniquely determines q(=q(ρ)) and λ(=λ(ρ)) defines a continuous function of ρ on (0,1).
Remark 3.5. We note that for λ>0 if u is a positive solution of (1.5) with ||u||∞=ρ∈(0,1), then there exists a unique x∗∈(0,1) such that u is symmetric about the point x∗, increasing on (0,x∗), decreasing on (x∗,1), u′(x∗)=0, and ||u||∞=u(x∗) where
x∗=∫ρ0dz√F(ρ)−F(z)∫ρ0dz√F(ρ)−F(z)+∫ρqdz√F(ρ)−F(z) |
and q=u(1) is uniquely determined by (3.7) (see Figure 10).
The next analytical result gives a complete description of the dynamics of (1.3) in the case when γ1→∞.
Theorem 3.6. Let γ2>0 be fixed. Then the following hold:
(1) if λ≤~E1(γ2) then (1.5) has no positive solution and the trivial solution is globally asymptotically stable;
(2) if λ>~E1(γ2) then (1.5) has a unique globally asymptotically stable positive solution uλ such that ‖uλ‖∞→1− as λ→∞ and ‖uλ‖∞→0 as λ→~E1(γ2)+. In fact, uλ→1− uniformly on compact subsets of (0,1) as λ→∞.
Recall, ~E1(γ2)=E1(∞,γ2)(=limγ1→∞E1(γ1,γ2)) is the principal eigenvalue of (2.6).
Figure 11 illustrates Theorem 3.6.
Remark 3.6. We show in the appendix that
~E1(γ2)=[π+arctan(−1γ2)]2. |
Note that for a fixed γ2>0,
1) ~E1(γ2)∈(π24,π2) for γ2>0;
2) ~E1(γ2)→π24 as γ2→0, and ~E1(γ2)→ED1=π2 as γ2→∞;
3) ~E1(γ2) is continuous and increasing in γ2.
The next result establishes that positive solutions of (1.5) have an ordering with respect to γ2 (see Figure 12).
Theorem 3.7. Let γ2>γ∗2>0 be fixed. If wλ:=uλ(γ2) and vλ:=uλ(γ∗2) are positive solutions of (1.5), then wλ<vλ; (0,1].
Finally, combining Theorems 3.3, 3.7 and Remarks 3.2, 3.6, we observe the limiting behavior of bifurcation curves of positive solutions for (1.4) as γ1→∞ in Figure 13a for a fixed γ2>0 and in Figure 13b for γ2→∞.
Following the methodology outlined in Section 3.1, we numerically obtain the evolution of bifurcation curves of positive solutions for (1.5) with respect to γ2 using MATLAB. Remark 3.6 now allows a characterization of MPS in this case:
Remark 3.7. We have the following:
1) limγ2→0+ℓ∗(∞,γ2)=π2√Dr=√ED14Dr(=ℓ∗(∞,0)say);
2) limγ2→∞ℓ∗(∞,γ2)=π√Dr=√ED1Dr(=ℓ∗(∞,∞)say).
Figure 14 shows an evolution of bifurcation curves as γ2 varies and r=D is chosen for convenience of presentation. Remark 3.7 is reflected in both Figure 14a where we observe shifting bifurcation curves to the left and Figure 14b where we observe shifting bifurcation curves to the right. Further, we observe that MPS (ℓ∗(∞,γ2)) ranges over (π2,π), increases in γ2, and causes bifurcation curves to move to the right as γ2 increases.
In Figure 15, we provide the evolution of MPS (ℓ∗(∞,γ2)) as γ2 varies with r=D chosen for convenience of presentation. Here, we observe that ℓ∗(∞,γ2) is an increasing function of γ2, approaches √ED14=π2 as γ2→0+, and approaches √ED1=π as γ2→∞, where ED1 is the principal eigenvalue of (2.4).
The model predicts that patches with size (ℓ) less than or equal to the minimum patch size, i.e., ℓ∗(γ1,γ2), cannot support a population as losses due to mortality caused by interaction with the hostile matrix fail to be overcome by growth inside the patch. However, for patches with ℓ>ℓ∗(γ1,γ2), the model predicts that organisms with a positive initial density profile will be able to successfully colonize the patch and persist. Our results also show that as patch size approaches ℓ∗(γ1,γ2) from the right, maximum density of the unique steady state density profile will tend towards zero in a continuous fashion. On the other hand, as patch size becomes large, a core habitat will form in the center of the patch for which organisms dwelling in the core are not likely to exhibit mortality at the patch/matrix interface. As patch size grows, this core habitat will approach 100% of the overall patch. Our results further show that the steady state density profile will approach 100% of patch carrying capacity (K) in this habitat core.
The composite parameter γi measures the overall effect of matrix hostility from interaction with ΩMi on population persistence as the MPS is an increasing function of γi. As Table 3 reveals, this effect is determined by the ratio of matrix death rate to patch intrinsic growth rate, habitat preference as measured by emigration probability (i.e., 1−αi, where αi is the probability that an organism will remain in the patch upon reaching the boundary), and in the Type Ⅱ DD scenario the ratio of patch diffusion rate to matrix diffusion rate. In all patch/matrix interface scenarios, increasing matrix death rate, increasing patch diffusion rate, decreasing patch intrinsic growth rate, or increasing emigration probability will cause an increase in MPS (holding all other factors fixed). In the Type Ⅱ DD scenario, increasing matrix diffusion rate will actually lower MPS. This suggests that for species with no difference between random walk step length in the matrix versus patch, decreasing movement resistance in the matrix can actually lower their MPS requirement. However, for organisms falling under the remaining scenarios, movement resistance in the matrix is not predicted to have any impact on MPS.
One matrix component's combined hostility measure remaining fixed as the other component's is varied (as was studied in Case 1 of our analysis) can arise naturally in a couple of different ways. Local matrix heterogeneity may occur in an obvious way as one matrix component is affected by consistent anthropogenic activities such as farming (e.g., increasing the matrix death rate in that component by excessive pesticide use), while the other component remains untouched. However, our results reveal that a second more stealthy situation may arise in which both matrix components are identical to the organism (e.g., matrix death rate and emigration probability are equal and fixed, while movement resistance in both matrix components is increasing in a similar fashion causing a reduction in matrix diffusion rate), but one matrix component is functionally different enough from the other to cause movement step lengths on one side to be the same in the matrix versus patch and different in the other. In this case, as matrix diffusion rate increases in both matrix components, combined hostility in the matrix component with similar movement step lengths between patch and matrix increases on that side, but no change in hostility occurs on the other side. These results indicate that even simple assumptions regarding movement at the individual-level could have lasting impacts at the patch-level via increasing of MPS requirement.
Also, MPS estimates in Case 1 of our analysis were shown to be confined to an envelope of possible values dictated by the fixed component's γi-value. For example, we considered the case when γ2=1 and combined matrix hostility was allowed to vary in ΩM1. Our results show that MPS ℓ∗(γ1,γ2)∈[ℓ∗(0,γ2),ℓ∗(∞,γ2)]). Although both lower and upper bounds on MPS are dependent on γ2, the length of this interval was always L0=π2√Dr–independent of γ2. This result sheds some light on the consequences of making an assumption of a homogeneous (or even locally homogeneous) matrix when modeling a population residing in a patch with locally heterogeneous matrix. In fact, we are able to employ this result to provide some bounds on the potential error incurred by assuming a locally homogeneous matrix. In our one-dimensional landscape, using the same γi-value for both matrix components yields a MPS prediction at worse L0 from the correct estimate.
When neither matrix component is immediately hostile to organisms, the model predicts that all positive steady state density profiles will be symmetric about the center of the patch if and only if the combined matrix hostility in each component is the same (i.e., composite parameters γ1 and γ2 are equal). Such a prediction arises from the fact that equal mortality at both patch/matrix interfaces create a symmetric spatial structure as shown in Figure 7b. This result captures what was found in [22,39] for the locally homogeneous matrix case. However, our results show that local heterogeneity in the matrix can create steady state density profiles with asymmetric spatial structure (see Figure 7a, c). In fact, peak density in a steady state profile is predicted to occur closer to the patch/matrix interface with lower combined hostility (e.g., γ1<γ2 implies that peak density of the steady state profile will occur closer to ΩM1). When one matrix component is close to being immediately hostile to organisms and the other not, the steady state density profile will always be skewed with the peak occurring closer to the other latter component. This fact can be used as a tool to give a quick evaluation of the presence of local matrix heterogeneity. For example, population densities empirically estimated from a patch that show such a skewed spatial structure where the peak density is closer to one part of the patch/matrix interface could indicate presence of local matrix heterogeneity and suggest the need for further investigation of matrix hostility, dispersal, and organismal habitat preference.
In this section, we present proofs of our main results. First, we provide proofs of Theorems 3.1–3.4 from Case 1.
Proof of Theorem 3.1: Here, we extend the study in [41], where a quadrature method was first introduced for the Dirichlet boundary conditions, and in [39] where it was extended for the case of linear boundary conditions with γ1=γ2.
Suppose u is a positive solution to (1.4) with ||u||∞=ρ. Clearly u″<0;(0,1). Further, the boundary conditions imply that u′(0)>0 and u′(1)<0. Hence, there exists a unique x∗∈(0,1) such that u′(x∗)=0 and ||u||∞=u(x∗)=ρ. Also, since the differential equation is autonomous, the solution must be symmetric about this x∗. Multiplying both sides of the differential equation in (1.4) by u′ and integrating we obtain
u′(x)={√2λ[F(ρ)−F(u(x))];(0,x∗]−√2λ[F(ρ)−F(u(x))];[x∗,1). | (4.1) |
Next, integrating (4.1) yields
∫u(x)q1dz√F(ρ)−F(z)=√2λx; x∈(0,x∗) | (4.2) |
∫u(x)q2dz√F(ρ)−F(z)=√2λ(1−x); x∈(x∗,1), | (4.3) |
where q1=u(0) and q2=u(1).
Now, taking x→x∗, λ, ρ, q1, q2, and x∗ must satisfy:
∫ρq1dz√F(ρ)−F(z)=x∗√2λ | (4.4) |
∫ρq2dz√F(ρ)−F(z)=(1−x∗)√2λ. | (4.5) |
From (4.4) and (4.5), we obtain
λ=12[∫ρq1dz√F(ρ)−F(z)+∫ρq2dz√F(ρ)−F(z)]2 | (4.6) |
and
x∗=∫ρq1dz√F(ρ)−F(z)∫ρq1dz√F(ρ)−F(z)+∫ρq2dz√F(ρ)−F(z). | (4.7) |
By the boundary conditions in (1.4) and by (4.1), we see that
2[F(ρ)−F(q1)]=γ21q21 and 2[F(ρ)−F(q2)]=γ22q22. | (4.8) |
Note here that since 2F(s)+γis2;i=1,2 are strictly increasing on (0,1), q1(=q1(ρ)) and q2(=q2(ρ)) are uniquely determined for a given ρ∈(0,1).
Next, let λ,q1,q2, and ρ satisfy (3.1) and (3.2). Define u:[0,1]→[0,ρ] such that u satisfies (4.2) for x∈(0,x∗) and (4.3) for x∈(x∗,1). Note that u is well-defined for x∈(0,x∗) since both ∫uq1dz√F(ρ)−F(z) and √2λx increase from 0 to ∫ρq1dz√F(ρ)−F(z) as u increases from q1 to ρ and x increases from 0 to x∗. Similarly we can see that u is well defined for x∈(x∗,1). Define H:(0,x∗)×(q1,ρ)→R by
H(l,v)=∫vq1dz√F(ρ)−F(z)−√2λl. |
Clearly, H is C1, H(x,u(x))=0 for x∈(0,x∗), and
Hv|(x,u(x))=1√F(ρ)−F(u(x))≠0. |
By the Implicit Function Theorem, u is C1 on (0,x∗). Similarly we can show that u is C1 on (x∗,1). Differentiating (4.2) and (4.3), we obtain
u′(x)={√2λ[F(ρ)−F(u(x));(0,x∗)−√2λ[F(ρ)−F(u(x));(x∗,1). | (4.9) |
Now, from (4.9), it is easy to show that u∈C2(0,1) and is a solution of −u″(x)=λf(u(x));(0,1). Further, u∈C1[0,1] and from (3.2) and (4.9), we obtain −u′(0)+√λγ1u(0)=0. Similarly, we see that u′(1)+√λγ2u(1)=0.
Finally, we recall the study in [41] where the corresponding λ (for the Dirichlet case q1=0=q2) was proved to be a well-defined and continuous function of ρ (using the fact that f(ρ)>0). Here q1 and q2, though not zero, are still continuous functions of ρ by (3.2). Hence, the arguments in [41] can be extended to show that λ in (3.1) is also well-defined and continuous for ρ on (0,1).
Proof of Theorem 3.2: First, when λ>E1(γ1,γ2) and λ≈E1(γ1,γ2), we establish existence of a positive solution uλ such that ||uλ||∞→0 as λ→E1(γ1,γ2)+. Let λ>E1(γ1,γ2), λ≈E1(γ1,γ2), and let σλ(γ1,γ2) be the principal eigenvalue of (2.2) and ϕλ be the corresponding normalized eigenfunction such that ϕλ>0; ¯Ω (see proof of Lemma 2.2 and Remark.1). Let ψλ=τϕλ, with τ>0. Then we have
−ψ″λ−λψλ(1−ψλ)=τϕλ(σλ(γ1,γ2)+λτϕλ)<0; Ω | (4.10) |
for τ≈0 since by Lemma 2.2 we have σλ(γ1,γ2)<0. It is also clear that ψλ satisfies the boundary conditions of (1.4). Thus, ψλ is a subsolution of (1.4) for τ≈0. Now, let Zλ=δλϕλ, where δλ=−σλ(γ1,γ2)λmin[0,1]ϕλ. We note that δλ>0 and δλ→0 as λ→E1(γ1,γ2)+ since σλ(γ1,γ2)<0, σλ(γ1,γ2)→0 as λ→E1(γ1,γ2)+, and minx∈[0,1]ϕλ↛0 as λ→E1(γ1,γ2)+. Then we have
−Z″λ−λZλ(1−Zλ)=δλ(λ+σλ(γ1,γ2))−(λδλϕλ)(1−δλϕλ)=δλϕλ(σλ(γ1,γ2)+λδλϕλ)≥0; Ω. |
It is also easy to see that Zλ satisfies the boundary conditions of (1.4). Thus, Zλ is a supersolution of (1.4) such that ||Zλ||∞→0+ as λ→E1(γ1,γ2)+.
Further, we have ψλ≤Zλ for τ≈0. By Lemma 2.1, (1.4) has a positive solution uλ∈[ψλ,Zλ] such that ||uλ||∞→0 as λ→E1(γ1,γ2)+. We emphasize that this shows existence of a positive solution for λ>E1(γ1,γ2) and λ≈E1(γ1,γ2).
It is well known that
{−v″=λv(1−v);(0,1)v(0)=0v(1)=0 |
has a unique positive solution vλ for λ>ED1 such that ||vλ||∞→1− as λ→∞. In fact, vλ→1− uniformly on compact subsets of Ω as λ→∞. Since v′λ(1)<0 and v′λ(0)>0, is it easy to see that vλ is a subsolution of (1.4) for λ≫1. Recall that Z≡1 is a supersolution of (1.4). By Lemma 2.1, (1.4) has a positive solution uλ∈[vλ,1] for λ>ED1. Furthermore, uλ→1− uniformly on compact subsets of Ω as λ→∞ since vλ does the same. This immediately implies that ||uλ||∞→1− as λ→∞. Our previous results together with Theorem 3.1 imply that (1.4) has a positive solution uλ for λ>E1(γ1,γ2) such that ||uλ||∞→0 as λ→E1(γ1,γ2)+ and ||uλ||∞→1 as λ→∞.
Next, we show that (1.4) has at most one positive solution for any λ>0. Suppose that (1.4) has two distinct positive solutions u1 and u2. (Note: Nontrivial, non-negative solutions are positive in the interior). Since Z≡m is a global supersolution for all m≥1, without loss of generality we can assume that u2 is the maximal positive solution of (1.4). Therefore, u1≤u2 in [0,1]. Integration by parts yields
∫10[u″1u2−u″2u1]dx=[u′1u2−u′2u1]|10=[u′1(1)u2(1)−u′2(1)u1(1)]−[u′1(0)u2(0)−u′2(0)u1(0)]=[−√λγ2u1(1)u2(1)+√λγ2u2(1)u1(1)]−[√λγ1u1(0)u2(0)−√λγ1u2(0)u1(0)]=0. |
However, by (1.4), we have
∫10[u″1u2−u″2u1]dx=∫10{(−λu1[1−u1])u2−(−λu2[1−u2])u1}dx=λ∫10u1u2[u1−u2]dx. |
Combining these results, it follows that
∫10u1u2[u1−u2]dx=0. |
This is a contradiction since u1 and u2 are distinct positive solutions with u1≤u2. Therefore, we must have u1≡u2. Hence, (1.4) has at most one positive solution uλ for λ>0.
Combining the above results that at most one positive solution exists for λ>0, existence of a positive solution uλ for λ>E1(γ1,γ2) and λ≈E1(γ1,γ2) such that ||uλ||∞→0 as λ→E1(γ1,γ2)+ and ||uλ||∞→1 as λ→∞, and the continuity of λ for ρ on (0,1) from Theorem 3.1, it follows that there is no positive solution for λ≤E1(γ1,γ2). Therefore, the exact bifurcation diagram illustrated in Figure 5 is established. Stability results immediately follow from uniqueness of positive steady states and the sub-supersolution construction [42].
Proof of Theorem 3.3: Let γ1>γ∗1>0 and λ>E1(γ1,γ2). Further, let wλ=uλ(γ1,γ2) be the positive solution of:
{−u″=λu(1−u);(0,1)−u′(0)+√λγ1u(0)=0u′(1)+√λγ2u(1)=0 | (4.11) |
and vλ=uλ(γ∗1,γ2) be the positive solution of:
{−u″=λu(1−u);(0,1)−u′(0)+√λγ∗1u(0)=0u′(1)+√λγ2u(1)=0. | (4.12) |
We claim that wλ is a subsolution of (4.12). To see this, observe that
−w″λ=λwλ(1−wλ);(0,1)−w′λ(0)+√λγ∗1wλ(0)<−w′λ(0)+√λγ1wλ(0)=0w′λ(1)+√λγ2wλ(1)=0, |
since γ1>γ∗1. Thus, wλ is a strict subsolution of (4.12). Note that Z≡1 is a supersolution of (4.12). By Lemma 2.1, (4.12) has a positive solution yλ satisfying wλ≤yλ≤1. Since wλ is a strict subsolution of (4.12), it is easy to see that wλ<yλ. However, since vλ is the unique solution of (4.12), we have yλ=vλ and hence wλ<vλ.
Proof of Theorem 3.4: First we show that if γ1>γ2, then q1<q2. Let uλ be a positive solution of (1.4) with ‖uλ‖∞=ρ, uλ(0)=q1, and uλ(1)=q2. Suppose that q1≥q2. Combining the equations in (3.2), we obtain 2[F(q1)−F(q2)]=γ22q22−γ21q21. Since F(u)=∫u0s(1−s)ds=12u2−13u3 is a strictly increasing function for u∈(0,1), we have F(q1)≥F(q2). This implies that γ22q22−γ21q21=(γ2q2+γ1q1)(γ2q2−γ1q1)≥0. Hence, we have γ2q2−γ1q1≥0, or equivalently, γ2γ1≥q1q2. This is a contradiction since q1≥q2 and γ1>γ2. Thus, we have q1<q2. Similarly we can show that if γ1<γ2, then q1>q2, and if γ1=γ2, then q1=q2.
Second, we provide proofs for Theorems 3.5–3.7 from Case 2.
Proof of Theorem 3.5: This is a special case of Theorem 3.1 where q1=0.
Proof of Theorem 3.6: First, we establish nonexistence of a positive solution when λ≤~E1(γ2). Suppose that uλ is a positive solution of (1.5) for λ≤~E1(γ2). Let ˜σλ(γ2) be the principal eigenvalue and ϕλ be the normalized positive eigenfunction of the eigenvalue problem (2.5) (see proofs of Lemma 2.3, 2.4 and Remark.2). Using integration by parts, we have
∫10[−ϕ″λuλ+u″λϕλ]dx={−ϕ′λuλ+u′λϕλ}|10={−ϕ′λ(1)uλ(1)+u′λ(1)ϕλ(1)}−{−ϕ′λ(0)uλ(0)+u′λ(0)ϕλ(0)}=√λγ2ϕ(1)uλ(1)−√λγ2uλ(1)ϕλ(1)=0. |
By (2.5), we also have
∫10[−ϕ″λuλ+u″λϕλ]dx=∫10{(λ+˜σλ(γ2))ϕλuλ−λuλ(1−uλ)ϕλ}dx=∫10ϕλuλ[˜σλ(γ2)+λuλ]dx>0 |
since ˜σλ(γ2)≥0 for λ≤~E1(γ2) by Lemma 2.3. This is a contradiction. Thus, (1.5) has no positive solution when λ≤~E1(γ2).
Next, we establish existence of a positive solution for λ>~E1(γ2). Let ψλ:=ωϕλ, where ω>0 is a constant to be chosen later. Then for x∈(0,1), we have ϕλ{˜σλ(γ2)+λωϕλ}≤0 for ω≈0 since ˜σλ(γ2)<0 for λ>~E1(γ2). This implies that
−ψ″λ=ω(λ+˜σλ(γ2))ϕλ≤λωϕλ(1−ωϕλ)=λψλ(1−ψλ). |
Also, on the boundary we have ψλ(0)=ωϕλ(0)=0 and ψ′(1)+√λγ2ψλ(1)=ωϕ′λ(1)+√λγ2ωϕλ(1)=0. Thus, ψλ is a subsolution of (1.5) for ω≈0. Further, it is clear that Z≡1 is a supersolution for (1.5). Hence by Lemma 2.1, (1.5) has a positive solution uλ∈[ψλ,Z] for λ>~E1(γ2).
We note that the proof of uniqueness of positive solutions for λ>~E1(γ2) is very similar to the proof of the uniqueness in Theorem 3.2. Hence we omit the proof here. Further, following the same argument as in Theorem 3.2, it follows that ||uλ||∞→1− as λ→∞. Finally, we show that ||uλ||∞→0 as λ→~E1(γ2)+. We note that Theorem 3.5 together with the facts that (1.5) has no positive solution for λ≤~E1(γ2) and has a unique positive solution uλ for λ>~E1(γ2) such that uλ→1− uniformly on compact subsets of Ω as λ→∞, immediately implying that ||uλ||∞→1− as λ→∞. These facts also establish that ||uλ||∞→0 as λ→~E1(γ2)+ since by Theorem 3.5, λ is a continuous function of ρ on (0,1). Hence, the exact bifurcation diagram illustrated in Figure 9 is established.
Proof of Theorem 3.7: This argument is similar to the proof of Theorem 3.3. Hence, we omit the proof.
Fragmentation and loss of natural habitats have driven a widespread decline in terrestrial biodiversity [16]. As demands for natural resources and land increase, landscapes will be increasingly altered and fragmented creating spatial heterogeneity [2]. Overall, effects of heterogeneity in landscapes and, particularly, matrix have been largely understudied compared with assessments based upon habitat amount or configuration [27]. Crucial to the area of conservation research is to what extent does loss of habitat area versus habitat fragmentation per se versus matrix heterogeneity contribute to population viability [43]. One of the most important conservation questions to be answered is: how much must be conserved for persistence of a population to be ensured [2]? This question is often addressed at the patch-level by determining the minimum patch size necessary to promote a viable population. Modeling studies suggest dependence of minimum patch size on several factors including reproduction rate, emigration rate, population genetics of the organism, and various stochastic factors [2]. However, it has become apparent that the "matrix matters" in prediction of population persistence [12,13]. A key guide for conservation is a better understanding of the role played by matrix quality in fragmented landscapes [13].
In this paper, we have extended an established reaction diffusion modeling framework to model effects of matrix heterogeneity in population persistence at the patch level. This new framework allows for predicting persistence of populations residing in a patch surrounded by a locally heterogenous matrix. Our results give the exact dynamics of a population exhibiting logistic growth, an unbiased random walk in the patch and matrix, habitat preference at the patch/matrix interface, and two functionally different matrix types for a one-dimensional landscape. We show existence of a minimum patch size, below which population persistence is not possible. This MPS can be estimated via empirically derived estimates of patch intrinsic growth rate and diffusion rate, habitat preference, and matrix death and diffusion rates. Mechanistic derivation of MPS allows us to analyze qualitative dependence of MPS on these parameters.
Recently, ecologists have begun focusing more and more attention to matrix habitat quality [44,45]. Our theoretical results provide additional support to the notion that local matrix heterogeneity can greatly change model predictions. Asymmetries not found in a locally homogeneous matrix can occur, giving rise to a steady state density profile with peak density closer to the matrix component with less severe combined matrix hostility. This result suggests a test of local matrix heterogeneity. Empirical estimates of density throughout a patch that show peak density near the center of the patch would suggest local matrix homogeneity for that patch. However, asymmetric spatial patterns in the density with a peak closer to one part of the patch/matrix interface would suggest presence of local matrix heterogeneity and suggest further investigation into the cause of the functional differences in matrix components. Our results also suggest that MPS estimates can be much less accurate when local heterogeneity is ignored in the modeling framework. For our one-dimensional landscape system, we have derived an upper bound on the MPS error in making such an assumption. Finally, our results agree with and strengthen previous authors' claims [2] that conservation strategies should not only consider patch size, configuration, and quality, but also quality and spatial structure of the surrounding matrix.
This work is supported by NSF grants Goddard (DMS-1853372) and Shivaji (DMS-1853352).
All authors declare no conflicts of interest in this paper.
Proof of Remark 3.2: First, we note that the eigenvalues need to be positive. Now the general solution of the differential equation in (2.1) for E>0 is ϕ(x)=c1cos(√Ex)+c2sin(√Ex), with c1,c2 constants to be determined. Using the boundary conditions, we obtain the following eigenvalue equation:
(γ1+γ2)cos(√E)+(γ1γ2−1)sin(√E)=0. | (A.1) |
Assuming γ1γ2≠1, this becomes
tan(√E)=γ1+γ21−γ1γ2. | (A.2) |
For γ1γ2>1, we know γ1+γ21−γ1γ2<0. Hence there exists a unique E∈(π24,π2) such that (A.2) is satisfied. Now, if γ1γ2<1, we know γ1γ21−γ1γ2>0. Hence there exists a unique E∈(0,π24) such that (A.2) is satisfied. In both cases, this shows the existence of the principal eigenvalue. From (A.2), we obtain the following explicit form of the principal eigenvalue:
E1(γ1,γ2)={[tan−1(γ1+γ21−γ1γ2)]2forγ1γ2<1[π+tan−1(γ1+γ21−γ1γ2)]2forγ1γ2>1. | (A.3) |
It follows from (A.1) and (A.3) that E1(γ1,γ2)→π24 as γ1→1γ2, E1(1γ2)=π24, and E1(⋅,γ2) is continuous and increasing in γ1.
Proof of Remark 3.6: Again, we first note that the eigenvalues need to be positive. Now for fixed γ2>0, the general solution of the differential equation in (2.6) for E>0 is ϕ(x)=c1cos(√Ex)+c2sin(√Ex), with c1,c2 constants to be determined. Using the boundary conditions, we obtain the following eigenvalue equation:
tan(√E)=−1γ2. | (A.4) |
Since tan(√E)∈(−∞,∞) and −1γ2<0, there exists a unique E∈(π24,π2) such that (A.4) is satisfied. This shows the existence of the principal eigenvalue. From (A.4), we obtain its explicit form:
~E1(γ2)=[π+tan−1(−1γ2)]2. | (A.5) |
From (A.5), it is easy to see that as γ2→0, ~E1(γ2)→π24 and as γ2→∞, ~E1(γ2)→π2. It is also easy to see that ~E1(γ2) is continuous and increasing in γ2.
Proof of Lemma 2.2: First, we consider the case γ1γ2≠1 and establish the existence and uniqueness of the principal eigenvalue σλ(γ1) by an application of the Implicit Function Theorem. Let λ>0 be fixed with λ+σ>0 in (2.2). Then the general solution of (2.2) has the form ϕλ(x)=c1cos(√λ+σx)+c2sin(√λ+σx), where c1 and c2 are constants. Using the boundary conditions and the fact that λ+σ>0, we obtain the following eigenvalue equation:
[(1−γ1γ2)λ+σ]tan(√λ+σ)=(γ1+γ2)√λ√λ+σ. | (A.6) |
Since γ1γ2≠1, we obtain an equivalent form of (A.6):
tan(√λ+σ)=−(γ1+γ2)√λ√λ+σ(γ1γ2−1)λ−σ. | (A.7) |
Using (A.7), we define
F(λ,σ):=tan(√λ+σ)+(γ1+γ2)√λ√λ+σ(γ1γ2−1)λ−σ. | (A.8) |
Note that F(E1(γ1,γ2),0)=0.
Next, we show that Fσ(E1(γ1,γ2),0)>0. A simple calculation will show that
Fσ(E1(γ1,γ2),0)=sec2(√E1(γ1,γ2))2√E1(γ1,γ2)+(γ1+γ2)√E1(γ1,γ2)(γ1γ2+1)2[(γ1γ2−1)E1(γ1,γ2)]2. | (A.9) |
By (A.9), it is easy to see that Fσ(E1(γ1,γ2),0)>0(≠0). By the Implicit Function Theorem, there exists ε>0 and a unique C1 function σ(λ)=σλ(γ1) defined on I:=(E1(γ1,γ2)−ε,E1(γ1,γ2)+ε) satisfying F(λ,σ(λ))=0 on I with F(E1(γ1,γ2),0)=0. This establishes the existence and uniqueness of the principal eigenvalue, σλ, of (2.2) as a root of (A.8).
Next, we show that Fλ(E1(γ1,γ2),0)>0. Observe that
Fλ(λ,σ)=sec2(√λ+σ)2√λ+σ+−σ(γ1+γ2)((γ1γ2−1)λ−σ)2√λ2+λσ[(γ1γ2−1)λ−σ]2. |
For λ=E1(γ1,γ2) and σ=0, we have
Fλ(E1(γ1,γ2),0)=sec2(√E1(γ1,γ2))2√E1(γ1,γ2)>0. |
Using the fact that F(λ,σ(λ))=0 on (E1(γ1,γ2)−ε,E1(γ1,γ2)+ε) and differentiating (A.8) with respect to λ, we have Fλ+Fσσ′(λ)=0, which implies σ′(λ)=−FλFσ. Since Fλ(E1(γ1,γ2),0)>0 and Fσ(E1(γ1,γ2),0)>0, it follows that σ′(E1(γ1,γ2))<0. This means σ(λ) is decreasing on (E1(γ1,γ2)−ε,E1(γ1,γ2)+ε). Therefore, we have σ(λ)<0 for λ>E1(γ1,γ2) and λ≈E1(γ1,γ2), and σ(λ)>0 for λ<E1(γ1,γ2) and λ≈E1(γ1,γ2). Further, it is easy to see that σ(λ)→0 as λ→E1(γ1,γ2).
Now we focus on the case when γ1γ2=1. We first show that σλ(γ1) exists and then establish that σλ(γ1)<0 for λ>E1(γ1,γ2) and λ≈E1(γ1,γ2) and σλ(γ1)→0 as λ→E1(γ1,γ2)+. Fix λ>E1(γ1,γ2) such that λ=π24+η where η>0 and η≈0. Setting γ1γ2=1 in (A.6), we obtain a new eigenvalue equation:
tan(√λ+σ)=[(γ1+γ2)√λ]√λ+σσ. | (A.10) |
Let g(σ)=tan(√π24+η+σ) and h(σ)=[(γ1+γ2)√π24+η]√π24+η+σσ, defined for σ∈[−π24−η,∞), with the exception of singularities. Note that g has its first singularity when √λ+σ=π2, or equivalently, when σ=−η. Also, since g is increasing on [−π24−η,−η) with g(σ)>0 and h is decreasing on [−π24−η,−η), with h(σ)<0, we know g(s)≠h(s) for σ∈[−π24−η,−η). This means no eigenvalue exists on this interval (see Figure A1).
Next, observe that g is increasing on (−η,0) and g(σ)→−∞ as σ→−η+. We also have h(−η)<0, h is decreasing on (−η,0) and h(σ)→−∞ as σ→0−. This implies that (A.10) has a unique minimum solution on (−η,0). Thus, (2.2) has a unique principal eigenvalue σλ(γ1,γ2)(<0). It is easy to see that σλ(γ1,γ2)→0 as λ→π24+. Thus, we have σλ(γ1,γ2)→0 as λ→E1(γ1,γ2)+. A similar argument will show that σλ(γ1)>0 for λ<E1(γ1,γ2) and λ≈E1(γ1,γ2) and σλ(γ1)→0 as λ→E1(γ1,γ2)−.
Remark.1. We note that the eigenfunction, ϕλ(x), corresponding to (2.2) is given by
ϕλ(x)=c1{cos(√λ+σλ(γ1)x)+√λγ1√λ+σλ(γ1)sin(√λ+σλ(γ1)x)}. |
Setting λ=E1(γ1,γ2) and σλ(γ1)=0, it follows that
ϕE1(x)=c1sin(√E1(γ1,γ2)x)(cot(√E1(γ1,γ2)x)+γ1). |
Using the value of E1(γ1,γ2) in the case γ1γ2≠1, it is not hard to see that ϕλ(x)>0 when λ=E1(γ1,γ2) and σλ(γ1)=0. By continuity, we have ϕλ>0 for λ≈E1(γ1,γ2).
Proof of Lemma 2.3: Denote β=β(μ) as the principal eigenvalue of (2.3) with corresponding eigenfunction ϕ, chosen such that ϕ(x)>0; Ω. We denote ϕμ as the derivative of ϕ with respect to μ for fixed x and use ′ to denote differentiation with respect to x for fixed μ. However, we denote β′(μ) as differentiation of β with respect to μ. Now, differentiating (2.3) with respect to μ yields
{−ϕ″μ=β′(μ)ϕ(μ)+β(μ)ϕμ;(0,1)ϕμ(0)=0ϕ′μ(1)+ϕ(μ)(1)+μϕμ(1)=0. | (A.11) |
Next, we calculate β′(μ) for any μ>0. By Green's Second Identity and the fact that ϕ(μ)(0)=ϕμ(0)=0, we have that
∫10[−ϕ″(μ)ϕμ+ϕ(μ)ϕ″μ]dx=−ϕ′(μ)(1)ϕμ(1)+ϕ(μ)(1)ϕ′μ(1). | (A.12) |
From (2.3) and (A.11), we obtain
∫10[−ϕ″(μ)ϕμ+ϕ(μ)ϕ″μ]dx=−(ϕ(μ)(1))2 | (A.13) |
and
∫10[−ϕ″(μ)ϕμ+ϕ(μ)ϕ″μ]dx=−β′(μ)∫10(ϕ(μ))2dx. | (A.14) |
Combining (A.13) and (A.14) gives
β′(μ)=(ϕ(μ)(1))2∫10(ϕ(μ))2dx>0. | (A.15) |
Thus, β(μ) is increasing in μ for μ>0.
Green's First Identity and (2.3) yields
∫10[−ϕ″(μ)ϕ(μ)]dx=∫10(ϕ′(μ))2dx+μ(ϕ(μ)(1))2. | (A.16) |
We also have
∫10[−ϕ″(μ)ϕ(μ)]dx=β(μ)∫10ϕ(μ)2dx. | (A.17) |
Combining (A.16) and (A.17), we have that
β(μ)∫10(ϕ(μ))2dx=∫10(ϕ′(μ))2dx+μ(ϕ(μ)(1))2. | (A.18) |
Now by (A.15) and (A.18) we obtain
β′(μ)=β(μ)μ−1μ∫10ϕ2(μ)dx∫10(ϕ′(μ))2dx | (A.19) |
which implies that
β′(μ)≤β(μ)μ | (A.20) |
for μ>0. Thus, β is a concave function of μ for μ>0.
Now let β(μ)=[m(μ)]2 in (2.3), where m>0. We note that the general solution of (2.3) has the form ϕ(x)=c1cos(mx)+c2sin(mx), where c1 and c2 are constants. Using the boundary conditions and the fact that m>0, we obtain the following eigenvalue equation:
mcos(m)+μsin(m)=0, | (A.21) |
or equivalently,
−μ=mcot(m) | (A.22) |
Since we are interested in the principal eigenvalue β(μ) for μ>0, we have m∈(π2,π). Define j(m)=mcot(m) for m∈(π2,π). It is easy to see that j is decreasing, continuous, j(π2)=0 and limm→πj(m)=−∞. Thus, for each μ>0, there exists a unique m∈(π2,π), or equivalently, β(μ)∈(π24,π2) which satisfies (A.22). This implies that limμ→∞β(μ)=ED1=π2.
Proof of Lemma 2.4: Observe in Figure A2 that ~E1(γ2) is the y-coordinate of the intersection of β(μ) and μ2γ22. Recall that π24<~E1(γ2)<π2 for γ2>0. Let λ>E1(γ2) be fixed. Observe μ2γ22=λ when μ=√λγ2. Note that β(√λγ2)=λ+~σλ(γ2). In Figure A2, we see that γ2√λ>γ2√~E1(γ2), therefore, β(γ2√λ)<λ. Then we have λ+˜σλ(γ2)<λ which implies ˜σλ(γ2)<0. Using a similar argument and the geometry of Figure A2 we can show that ˜σλ(γ2)>0 when λ<~E1(γ2) and ˜σλ(γ2)→0 as λ→~E1(γ2).
Remark.2. The eigenfunction corresponding to (2.5) is ϕλ(x)=sin(√λ+˜σλ(γ2)x). By Figure A2 we have π2<√~E1(γ2)<√λ+˜σλ(γ2)<π. For x∈(0,1), this implies 0<√λ+˜σλ(γ2)x<π. On the interval (0,π), the sine function is positive, hence ϕλ>0.
[1] | E. O. Wilson, Threats to biodiversity, Sci. Am., 261 (1989), 108–117. http://www.jstor.org/stable/24987402 |
[2] |
L. Fahrig, How much habitat is enough?, Biol. Conserv., 100 (2001), 65–74. https://doi.org/10.1016/S0006-3207(00)00208-1 doi: 10.1016/S0006-3207(00)00208-1
![]() |
[3] |
I. Hanski, Habitat loss, the dynamics of biodiversity, and a perspective on conservation, AMBIO, 40 (2011), 248–255. https://doi.org/10.1007/s13280-011-0147-3 doi: 10.1007/s13280-011-0147-3
![]() |
[4] |
D. Tilman, M. Clark, D. R. Williams, K. Kimmel, S. Polasky, C. Packer, Future threats to biodiversity and pathways to their prevention, Nature, 546 (2017), 73–81. https://doi.org/10.1038/nature22900 doi: 10.1038/nature22900
![]() |
[5] |
K. J. J. Kuipers, J. P. Hilbers, J. Garcia-Ulloa, B. J. Graae, R. May, F. Verones, et al., Habitat fragmentation amplifies threats from habitat loss to mammal diversity across the world's terrestrial ecoregions, One Earth, 4 (2021), 1505–1513. https://doi.org/10.1016/j.oneear.2021.09.005 doi: 10.1016/j.oneear.2021.09.005
![]() |
[6] | J. F. Brodie, W. D. Newmark, Heterogeneous matrix habitat drives species occurrences in complex, fragmented landscapes, Am. Nat., 193 (2019), 748–754. https://www.journals.uchicago.edu/doi/abs/10.1086/702589 |
[7] |
M. A. Bowers, S. F. Matter, Landscape ecology of mammals: Relationships between density and patch size, J. Mammal., 78 (1997), 999–1013. https://doi.org/10.2307/1383044 doi: 10.2307/1383044
![]() |
[8] |
D. J. Bender, T. A. Contreras, L. Fahrig, Habitat loss and population decline: A meta-analysis of the patch size effect, Ecology, 79 (1998), 517–533. https://doi.org/10.1890/0012-9658(1998)079[0517:HLAPDA]2.0.CO;2 doi: 10.1890/0012-9658(1998)079[0517:HLAPDA]2.0.CO;2
![]() |
[9] |
E. F. Connor, A. C. Courtney, J. M. Yoder, Individuals–area relationships: The relationship between animal population density and area, Ecology, 81 (2000), 734–748, https://doi.org/10.1890/0012-9658(2000)081[0734:IARTRB]2.0.CO;2 doi: 10.1890/0012-9658(2000)081[0734:IARTRB]2.0.CO;2
![]() |
[10] |
P. A. Hamback, G. Englund, Patch area, population density and the scaling of migration rates: the resource concentration hypothesis revisited, Ecol. Lett., 8 (2005), 1057–1065. https://doi.org/10.1111/j.1461-0248.2005.00811.x doi: 10.1111/j.1461-0248.2005.00811.x
![]() |
[11] |
P. A. Hamback, M. Vogt, T. Tscharntke, C. Thies, G. Englund, Top-down and bottom-up effects on the spatiotemporal dynamics of cereal aphids: Testing scaling theory for local density, Oikos, 116 (2007), 1995–2006. https://doi.org/10.1111/j.2007.0030-1299.15800.x doi: 10.1111/j.2007.0030-1299.15800.x
![]() |
[12] |
T. H. Ricketts, The matrix matters: Effective isolation in fragmented landscapes, Am. Nat., 158 (2001), 87–99. https://doi.org/10.1086/320863 doi: 10.1086/320863
![]() |
[13] |
J. A. Prevedello, M. V. Vieira, Does the type of matrix matter? a quantitative review of the evidence, Biodiversity Conserv., 19 (2010), 1205–1223. https://doi.org/10.1007/s10531-009-9750-z doi: 10.1007/s10531-009-9750-z
![]() |
[14] |
J. T. Cronin, Matrix heterogeneity and host–parasitoid interactions in space, Ecology, 84 (2003), 1506–1516. http://dx.doi.org/10.1890/0012-9658(2003)084[1506:MHAHII]2.0.CO;2 doi: 10.1890/0012-9658(2003)084[1506:MHAHII]2.0.CO;2
![]() |
[15] |
J. T. Cronin, From population sources to sieves: the matrix alters host-parasitoid source-sink structure, Ecology, 88 (2007), 2966–2976. https://doi.org/10.1890/07-0070.1 doi: 10.1890/07-0070.1
![]() |
[16] |
B. T. Klingbeil, M. R. Willig, Matrix composition and landscape heterogeneity structure multiple dimensions of biodiversity in temperate forest birds, Biodiversity Conserv., 25 (2016), 2687–2708. https://doi.org/10.1007/s10531-016-1195-6 doi: 10.1007/s10531-016-1195-6
![]() |
[17] |
W. F. Fagan, R. S. Cantrell, C. Cosner, How habitat edges change species interactions, Am. Nat., 153 (1999), 165–182. https://doi.org/10.1086/303162 doi: 10.1086/303162
![]() |
[18] |
G. A. Maciel, F. Lutscher, How individual movement response to habitat edges affects population persistence and spatial spread, Am. Nat., 182 (2013), 42–52. https://doi.org/10.1086/670661 doi: 10.1086/670661
![]() |
[19] |
D. Ludwig, D. D. Jones, C. S. Holling, Qualitative analysis of insect outbreak systems: The spruce budworm and forest, J. Anim. Ecol., 47 (1978), 315–332. https://doi.org/10.2307/3939 doi: 10.2307/3939
![]() |
[20] |
O. Ovaskainen, S. J. Cornell, Biased movement at a boundary and conditional occupancy times for diffusion processes, J. Appl. Probab., 40 (2003), 557–580. https://doi.org/10.1239/jap/1059060888 doi: 10.1239/jap/1059060888
![]() |
[21] |
J. T. Cronin, J. Goddard Ⅱ, R. Shivaji, Effects of patch matrix-composition and individual movement response on population persistence at the patch-level, Bull. Math. Biol., 81 (2019), 3933–3975. https://doi.org/10.1007/s11538-019-00634-9 doi: 10.1007/s11538-019-00634-9
![]() |
[22] |
J. T. Cronin, N. Fonseka, J. Goddard Ⅱ, J. Leonard, R. Shivaji, Modeling the effects of density dependent emigration, weak allee effects, and matrix hostility on patch-level population persistence, Math. Biosci. Eng., 17 (2019), 1718–1742. https://doi.org/10.3934/mbe.2020090 doi: 10.3934/mbe.2020090
![]() |
[23] |
J. Goddard Ⅱ, Q. Morris, C. Payne, R. Shivaji, A diffusive logistic equation with u-shaped density dependent dispersal on the boundary, Topol. Methods Nonlinear Anal., 53 (2019), 335–349. https://doi.org/10.12775/TMNA.2018.047 doi: 10.12775/TMNA.2018.047
![]() |
[24] |
N. Fonseka, J. Goddard Ⅱ, Q. Morris, R. Shivaji, B. Son, On the effects of the exterior matrix hostility and a u-shaped density dependent dispersal on a diffusive logistic growth model, Discrete Contin. Dyn. Syst. Ser. B, 13 (2020), 3401–3415. http://dx.doi.org/10.3934/dcdss.2020245 doi: 10.3934/dcdss.2020245
![]() |
[25] |
J. T. Cronin, J. Goddard Ⅱ, A. Muthunayake, R. Shivaji, Modeling the effects of trait-mediated dispersal on coexistence of mutualists, Math. Biosci. Eng., 17 (2020), 7838–7861. https://doi.org/10.3934/MBE.2020399 doi: 10.3934/MBE.2020399
![]() |
[26] |
N. Fonseka, J. Machado, R. Shivaji, A study of logistic growth models influenced by the exterior matrix hostility and grazing in an interior patch, Electron. J. Qual. Theory Diff. Equations, 2020 (2020), 1–11. https://doi.org/10.14232/ejqtde.2020.1.17 doi: 10.14232/ejqtde.2020.1.17
![]() |
[27] |
L. Fahrig, J. Baudry, L. Brotons, F. G. Burel, T. O. Crist, R. J. Fuller, et al., Functional landscape heterogeneity and animal biodiversity in agricultural landscapes, Ecol. Lett., 14 (2011), 101–112. https://doi.org/10.1111/j.1461-0248.2010.01559.x doi: 10.1111/j.1461-0248.2010.01559.x
![]() |
[28] |
S. A. Levin, Dispersion and population interactions, Am. Nat., 108 (1974), 207–228. https://doi.org/10.1086/282900 doi: 10.1086/282900
![]() |
[29] |
S. A. Levin, The role of theoretical ecology in the description and understanding of populations in heterogeneous environments, Am. Zool., 21 (1981), 865–875. https://doi.org/10.1093/icb/21.4.865 doi: 10.1093/icb/21.4.865
![]() |
[30] | P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Springer-Verlag, 1979. |
[31] | A. Okubo, Diffusion and Ecological Problems: Mathematical Models, Springer, Berlin, 1980. |
[32] | J. D. Murray, Mathematical Biology. II, 3rd edition, Springer-Verlag, New York, 2003. |
[33] | R. S. Cantrell, C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley, Chichester, 2003. |
[34] |
E. E. Holmes, M. A. Lewis, R. R. V. Banks, Partial differential equations in ecology: spatial interactions and population dynamics, Ecology, 75 (1994), 17–29. https://doi.org/10.2307/1939378 doi: 10.2307/1939378
![]() |
[35] |
O. Ovaskainen, Habitat-specific movement parameters estimated using mark–recapture data and a diffusion model, Ecology, 85 (2004), 242–257. https://doi.org/10.1890/02-0706 doi: 10.1890/02-0706
![]() |
[36] |
D. Ludwig, D. G. Aronson, H. F. Weinberger, Spatial patterning of the spruce budworm, J. Math. Biol., 8 (1979), 217–258. https://doi.org/10.1007/BF00276310 doi: 10.1007/BF00276310
![]() |
[37] |
R. S. Cantrell, C. Cosner, Diffusion models for population dynamics incorporating individual behavior at boundaries: Applications to refuge design, Theor. Popul. Biol., 55 (1999), 189–207. https://doi.org/10.1006/tpbi.1998.1397 doi: 10.1006/tpbi.1998.1397
![]() |
[38] |
R. S. Cantrell, C. Cosner, Density dependent behavior at habitat boundaries and the allee effect, Bull. Math. Biol., 69 (2007), 2339–2360. https://doi.org/10.1007/s11538-007-9222-0 doi: 10.1007/s11538-007-9222-0
![]() |
[39] |
J. Goddard Ⅱ, Q. Morris, S. Robinson, R. Shivaji, An exact bifurcation diagram for a reaction diffusion equation arising in population dynamics, Boundary Value Prob., 170 (2018), 1–17. https://doi.org/10.1186/s13661-018-1090-z doi: 10.1186/s13661-018-1090-z
![]() |
[40] |
H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered banach spaces, SIAM Rev., 18 (1976), 620–709. https://doi.org/10.1137/1018114 doi: 10.1137/1018114
![]() |
[41] | T. Laetsch, The number of solutions of a nonlinear two point boundary value problem, Indiana Univ. Math. J., 20 (1970), 1–13. http://www.jstor.org/stable/24890103 |
[42] | C. V. Pao, Nonlinear parabolic and elliptic equations, Plenum Press, New York, 1992. |
[43] |
D. J. Bruggeman, T. Wiegand, N. Fernández, The relative effects of habitat loss and fragmentation on population genetic variation in the red-cockaded woodpecker (picoides borealis), Mol. Ecol., 19 (2010), 3679–3691. https://doi.org/10.1111/j.1365-294X.2010.04659.x doi: 10.1111/j.1365-294X.2010.04659.x
![]() |
[44] |
E. E. Crone, L. M. Brown, J. A. Hodgson, F. Lutscher, C. B. Schultz, Faster movement in nonhabitat matrix promotes range shifts in heterogeneous landscapes, Ecology, 100 (2019), 1–10. https://doi.org/10.1002/ecy.2701 doi: 10.1002/ecy.2701
![]() |
[45] |
J. S. MacDonald, F. Lutscher, Individual behavior at habitat edges may help populations persist in moving habitats, J. Math. Biol., 77 (2018), 2049–2077. https://doi.org/10.1007/s00285-018-1244-8 doi: 10.1007/s00285-018-1244-8
![]() |
Scenario Name | Scenario Description | κi | References |
Continuous Density | Organisms move between the patch and matrix with equal probability. Step sizes and movement probabilities are equal in the patch and matrix. | 1 | [36] |
Type Ⅰ Discontinuous Density (DD) | Organisms modify their movement behavior at the patch/matrix interface and would have a probability α of remaining in ˜Ω which may be different from 50%. Step sizes differ between the patch and matrix, whereas movement probabilities are equal. | √D0iD | [18,20] |
Type Ⅱ Discontinuous Density (DD) | Organisms modify their movement behavior at the patch/matrix interface and would have a probability α of remaining in ˜Ω which may be different from 50%. Step sizes are equal between the patch and matrix but movement probabilities are different. | D0iD | [18,20] |
Type Ⅲ Discontinuous Density (DD) | Organisms remain in ˜Ω with probability α which may be different from 50%. Movement probabilities and step sizes are the same between the patch and matrix. | 1 | [37,38] |
Parameter | Definition | Units |
ℓ | Length of patch ˜Ω | length |
r | Patch intrinsic growth rate | 1/time |
D | Patch diffusion rate | length2/time |
K | Patch carrying capacity | density |
S0i | Death rate in matrix component ˜ΩMi | 1/time |
D0i | Diffusion rate in matrix component ˜ΩMi | length2/time |
κi | Patch/matrix interface parameter for ˜ΩMi | unitless |
Composite Parameter | Expression | Range |
λ | rℓ2D | 0<λ<∞ |
CTS, Type Ⅰ DD, & Type Ⅲ DD: γi | √S0ir1−αiαi | 0≤γi<∞ |
Type Ⅱ DD: γi | √S0iDrD0i1−αiαi | 0≤γi<∞ |
Scenario Name | Scenario Description | κi | References |
Continuous Density | Organisms move between the patch and matrix with equal probability. Step sizes and movement probabilities are equal in the patch and matrix. | 1 | [36] |
Type Ⅰ Discontinuous Density (DD) | Organisms modify their movement behavior at the patch/matrix interface and would have a probability α of remaining in ˜Ω which may be different from 50%. Step sizes differ between the patch and matrix, whereas movement probabilities are equal. | √D0iD | [18,20] |
Type Ⅱ Discontinuous Density (DD) | Organisms modify their movement behavior at the patch/matrix interface and would have a probability α of remaining in ˜Ω which may be different from 50%. Step sizes are equal between the patch and matrix but movement probabilities are different. | D0iD | [18,20] |
Type Ⅲ Discontinuous Density (DD) | Organisms remain in ˜Ω with probability α which may be different from 50%. Movement probabilities and step sizes are the same between the patch and matrix. | 1 | [37,38] |
Parameter | Definition | Units |
ℓ | Length of patch ˜Ω | length |
r | Patch intrinsic growth rate | 1/time |
D | Patch diffusion rate | length2/time |
K | Patch carrying capacity | density |
S0i | Death rate in matrix component ˜ΩMi | 1/time |
D0i | Diffusion rate in matrix component ˜ΩMi | length2/time |
κi | Patch/matrix interface parameter for ˜ΩMi | unitless |
Composite Parameter | Expression | Range |
λ | rℓ2D | 0<λ<∞ |
CTS, Type Ⅰ DD, & Type Ⅲ DD: γi | √S0ir1−αiαi | 0≤γi<∞ |
Type Ⅱ DD: γi | √S0iDrD0i1−αiαi | 0≤γi<∞ |