Most of heterogeneous epidemic models assume exponentially distributed sojourn times in infectious states, which may not be practical in reality and could affect the dynamics of the epidemic. This paper investigates the potential discrepancies between exponential and non-exponential distribution models in analyzing the transmission patterns of infectious diseases and evaluating control measures. Two SEIHR models with multiple subgroups based on different assumptions for latency are established: Model Ⅰ assumes an exponential distribution of latency, while Model Ⅱ assumes a gamma distribution. To overcome the challenges associated with the high dimensionality of GDM, we derive the basic reproduction number (R0) of the model theoretically, and apply numerical simulations to evaluate the effect of different interventions on EDM and GDM. Our results show that considering a more realistic gamma distribution of latency can change the peak numbers of infected and the timescales of an epidemic, and GDM may underestimate the infection eradication time and overestimate the peak value compared to EDM. Additionally, the two models can produce inconsistent predictions in estimating the time to reach the peak. Our study contributes to a more accurate understanding of disease transmission patterns, which is crucial for effective disease control and prevention.
Citation: Huiping Zang, Shengqiang Liu, Yi Lin. Evaluations of heterogeneous epidemic models with exponential and non-exponential distributions for latent period: the Case of COVID-19[J]. Mathematical Biosciences and Engineering, 2023, 20(7): 12579-12598. doi: 10.3934/mbe.2023560
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[2] | Yongli Cai, Malay Banerjee, Yun Kang, Weiming Wang . Spatiotemporal complexity in a predator--prey model with weak Allee effects. Mathematical Biosciences and Engineering, 2014, 11(6): 1247-1274. doi: 10.3934/mbe.2014.11.1247 |
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[6] | Tingfu Feng, Leyun Wu . Global dynamics and pattern formation for predator-prey system with density-dependent motion. Mathematical Biosciences and Engineering, 2023, 20(2): 2296-2320. doi: 10.3934/mbe.2023108 |
[7] | Jin Zhong, Yue Xia, Lijuan Chen, Fengde Chen . Dynamical analysis of a predator-prey system with fear-induced dispersal between patches. Mathematical Biosciences and Engineering, 2025, 22(5): 1159-1184. doi: 10.3934/mbe.2025042 |
[8] | Ilse Domínguez-Alemán, Itzel Domínguez-Alemán, Juan Carlos Hernández-Gómez, Francisco J. Ariza-Hernández . A predator-prey fractional model with disease in the prey species. Mathematical Biosciences and Engineering, 2024, 21(3): 3713-3741. doi: 10.3934/mbe.2024164 |
[9] | Muhammad Shoaib Arif, Kamaleldin Abodayeh, Asad Ejaz . On the stability of the diffusive and non-diffusive predator-prey system with consuming resources and disease in prey species. Mathematical Biosciences and Engineering, 2023, 20(3): 5066-5093. doi: 10.3934/mbe.2023235 |
[10] | Yuanshi Wang, Donald L. DeAngelis . A mutualism-parasitism system modeling host and parasite with mutualism at low density. Mathematical Biosciences and Engineering, 2012, 9(2): 431-444. doi: 10.3934/mbe.2012.9.431 |
Most of heterogeneous epidemic models assume exponentially distributed sojourn times in infectious states, which may not be practical in reality and could affect the dynamics of the epidemic. This paper investigates the potential discrepancies between exponential and non-exponential distribution models in analyzing the transmission patterns of infectious diseases and evaluating control measures. Two SEIHR models with multiple subgroups based on different assumptions for latency are established: Model Ⅰ assumes an exponential distribution of latency, while Model Ⅱ assumes a gamma distribution. To overcome the challenges associated with the high dimensionality of GDM, we derive the basic reproduction number (R0) of the model theoretically, and apply numerical simulations to evaluate the effect of different interventions on EDM and GDM. Our results show that considering a more realistic gamma distribution of latency can change the peak numbers of infected and the timescales of an epidemic, and GDM may underestimate the infection eradication time and overestimate the peak value compared to EDM. Additionally, the two models can produce inconsistent predictions in estimating the time to reach the peak. Our study contributes to a more accurate understanding of disease transmission patterns, which is crucial for effective disease control and prevention.
In 1987, Karevia and Odell [1] first proposed the following one-predator and one-prey model with prey-taxis in order to explain that an area-restricted search creates the following predator aggregation phenomenon
{ut=∇⋅(d(w)∇u)−∇⋅(uχ(w)∇w)+G1(u,w),wt=DΔw+G2(u,w), | (1.1) |
where D>0 is the diffusivity coefficient of preys, d(w) denotes the motility function of predators, χ(w) represents the prey-taxis sensitivity coefficient, and the term −∇⋅(uχ(w)∇w) stands for the tendency of the predator moving towards the increasing direction of the prey gradient, and it is viewed as the prey-taxis term. The functions G1(u,w) and G2(u,w) describe the predator-prey interactions, which include both intra-specific and inter-specific interactions. Generally the predator-prey interaction functions G1(u,w) and G2(u,w) possess the following prototypical forms.
G1(u,w)=γuF(w)−uh(u),G2(u,w)=−uF(w)+f(w), | (1.2) |
where γ>0 denotes the intrinsic predation rate, uF(w) represents the inter-specific interaction and uh(u) and f(w) stand for the intra-specific interaction. Specifically, F(w) is the functional response function accounting for the intake rate of predators as a function of prey density; it is often used in the following form in the literature [2,3,4]
F(w)=w(Holling type I),F(w)=wλ+w(Holling type II),F(w)=wmλm+wm(Holling type III),F(w)=1−e−λw(Ivlev type) | (1.3) |
with constants λ>0 and m>1; other types of functional response functions (e.g., Beddington-DeAngelis type in [5], Crowley-Martin type in [6]) and more predator-prey interactions can be found in [7,8,9,10]. The predator mortality rate function h(u) is typically of the form
h(u)=θ+αu, | (1.4) |
where θ>0 accounts for the natural death rate and α≥0 denotes the rate of death resulting from the intra-specific competition, which is also called the density-dependent death [11]. The prey growth function f(w) is usually assumed to be negative for large w due to the limitation of resources (or crowding effect), and its typical forms are
f(w)=μw(1−wK)(Logistic type),orf(w)=μw(1−wK)(wk−1)(Bistable or Allee effect type), | (1.5) |
where μ>0 is the intrinsic growth rate of prey, K>0 is called the carrying capacity and 0<k<K. Now there exist many interesting results about global existence, uniform boundedness, asymptotic behavior, traveling waves and pattern formation of solutions to System (1.1) or its variants in [11,12,13,14,15]. When d(w)=d>0 and χ(w)=χ>0, G1(u,v) and G2(u,v) have the forms of (1.2), Wu et al. [15] obtained the global existence and uniform persistence of solutions to (1.1) in any dimension provided that χ is suitably small. Then, Jin and Wang[13] derived the global boundedness and asymptotic stability of solutions for System (1.1) without the smallness assumption on χ in a two-dimensional bounded domain. Moreover, Wang et al. [16] studied the nonconstant positive steady states and pattern formation of (1.1) in a one-dimensional bounded domain. Under the conditions that d(w) and χ(w) are not constants and h(u) is given by (1.4), Jin and Wang[14] established the global boundedness, asymptotic behavior and spatio-temporal patterns of solutions for (1.1) under some conditions on the parameters in a two-dimensional smooth bounded domain. For more related results in predator-prey models, we refer the readers to [17,18,19,20,21,22,23,24,25,26,27,28,29] and the references therein.
However, all the aforementioned works are devoted to studying prey-taxis models with one-predator and one prey. Now let us mention some predator-prey models with two-predator and one-prey. Recently, the following general two-predator and one-prey model with prey-taxis has attracted a lot of attention.
{ut=∇⋅(d1(w)∇u)−∇⋅(uχ1(w)∇w)+γ1uF1(w)−uh1(u)−β1uv,vt=∇⋅(d2(w)∇v)−∇⋅(vχ2(w)∇w)+γ2vF2(w)−vh2(v)−β2uv,wt=DΔw−uF1(w)−vF2(w)+f(w), | (1.6) |
as applied in a smooth bounded domain Ω⊂Rn, n≥1. Given d1(w)=d2(w)=1 and β1=β2=0, Wang et al. [30] derived the global boundedness, nonconstant positive steady states and time-periodic patterns of solutions for System (1.6). Wang and Wang [31] studied the uniform boundedness and asymptotic stability of nonnegative spatially homogeneous equilibria for (1.6) in any dimension. Given d1(w)=d2(w)=1, χi(w)=χi>0(i=1,2) and β1=β2=β>0, Mi et al. [32] obtained the global boundedness and stability of classical solutions in any dimension under suitable conditions of parameters. Under the conditions that d1(w) and d2(w) are non-constants and β1=β2=0, Qiu et al.[33] rigorously proved the global existence, uniform boundedness and stabilization of classical solutions in any dimension with suitable conditions on motility functions and the coefficients of the logistic source. However, when β1,β2≠0, the global existence and stabilization of solutions for (1.6) are still open. Given χi(w)=−d′i(w)≥0 if d′i(w)≤0(i=1,2), the diffusion-advection terms in (1.6) can respectively become the forms Δ(d1(w)u) and Δ(d2(w)v), which could be interpreted as "density-suppressed motility" in [34,35]. This means that the predator will reduce its motility when encountering the prey, which is a rather reasonable assumption that has very sound applications in the predator-prey systems. Since the possible degeneracy caused by the density-suppressed motility brings considerable challenges for mathematical analysis, many works have showed various interesting results, which can be found in [36,37,38,39,40,41,42,43,44]. Given χi(w)=−d′i(w)≥0(i=1,2), F1(w)=F2(w)=w, h1(u)=u,h2(v)=v and f(w)=μw(m(x)−w), System (1.6) can be simplified as
{ut=Δ(d1(w)u)+u(γ1w−u−β1v),vt=Δ(d2(w)v)+v(γ2w−v−β2u),wt=DΔw−(u+v)w+μw(m(x)−w), | (1.7) |
where the parameters D,μ,γi,βi(i=1,2) are positive, the dispersal rate functions di(w)(i=1,2) satisfy following the hypothesis: di(w)∈C2([0,∞)),d′i(w)≤0 on [0,∞) and di(w)>0. Wang and Xu [45] have found some interesting results for System (1.7) in a two-dimensional smooth bounded domain. More specifically, when D=1 and m(x)=1, System (1.7) has a unique globally bounded classical solution. By constructing appropriate Lyapunov functionals and using LaSalle's invariant principle, the authors proved that the global bounded solution of (1.7) converges to the co-existence steady state exponentially or competitive exclusion steady state algebraically as time tends to infinity in different parameter regimes. For a prey's resource that is spatially heterogeneous (i.e., m(x) is non-constant), the authors used numerical simulations to demonstrate that the striking phenomenon "slower diffuser always prevails" given in [46,47] fails to appear if the non-random dispersal strategy is employed by competing species (i.e., either d1(w) or d2(w) is non-constant) while it still holds if both d1(w) and d2(w) are constants. However, there are few results about global boundedness and large time behavior of solutions for (1.7) in the general form.
Inspired by the above works, this paper is concerned with the following two-species competitive predator-prey system with the following density-dependent diffusion
{ut=Δ(d1(w)u)+γ1uF1(w)−uh1(u)−β1uv,(x,t)∈Ω×(0,∞),vt=Δ(d2(w)v)+γ2vF2(w)−vh2(v)−β2uv,(x,t)∈Ω×(0,∞),wt=DΔw−uF1(w)−vF2(w)+f(w),(x,t)∈Ω×(0,∞),∂u∂ν=∂v∂ν=∂w∂ν=0,(x,t)∈∂Ω×(0,∞),(u,v,w)(x,0)=(u0,v0,w0)(x),x∈Ω, | (1.8) |
where Ω⊂R2 is a bounded domain with a smooth boundary ∂Ω, ∂∂ν denotes the derivative with respect to the outward normal vector ν of ∂Ω, and the parameters D and γi, βi (i=1,2) are positive. The unknown functions u=u(x,t) and v=v(x,t) denote the densities of two-competing species (e.g., predators), and w=w(x,t) represents the density of predators' resources (e.g., the prey) at a position x and time t>0. When d1(w)=d1>0 and d2(w)=d2>0, System (1.8) becomes the well-known diffusive predator-prey system, which has been widely studied in [48,49,50,51]. However, to the best of our knowledge, the results of the two-predator and one-prey system given by (1.8) with density-suppressed motility (i.e., d1(w) and d2(w) are non-constants) indicate that the competition and general predator mortality rate hi(u) are almost vacant. The main aim of this paper is to explore the influence of the predation interaction, competition and general predator mortality on the dynamical behavior of System (1.8). Throughout this paper, we assume that the functions di(s),Fi(s),hi(s),(i=1,2),f(s) and initial data (u0,v0,w0) mentioned in (1.8) satisfy the following hypotheses:
(H1) di(s)∈C2([0,∞)) with di(s)>0 and d′i(s)≤0, i=1,2 on [0,∞). The typical example is di(s)=1(1+s)κi or di(s)=exp(−κis) with κi>0, i=1,2.
(H2) Fi(s)∈C1([0,∞)), Fi(0)=0, Fi(s)>0 and F′i(s)>0, i=1,2 in (0,∞).
(H3) hi(s)∈C2([0,∞)) and there exist constants θi>0 and αi≥0 such that hi(s)≥θi and h′i(s)≥αi, i=1,2 for all s>0.
(H4) f(s)∈C1([0,∞)) with f(0)=0, and there exist positive constants μ and K such that f(s)≤μs for all s≥0, f(K)=0 and f(s)<0 for s>K.
(H5) (u0,v0,w0)∈(W1,p(Ω))3 with p>2 and u0,v0,w0≥0.
Here, we note that there are many candidates for the above functions Fi(s), hi(s) and f(s) as in (1.3)–(1.5). Due to the presence of the prey's density dependent diffusion coefficient, Model (1.8) is a cross-diffusion system and the parabolic comparison principle is no longer applicable. Moreover, when α1=α2=0, the key L2-spatiotemporal estimates of u and v cannot be directly derived; thus, the uniform boundedness of solutions is not an obvious result and needs to be justified. Based on the above hypotheses, the first main result of this paper asserts the global existence and boundedness of solutions for System (1.8) as follows.
Theorem 1.1. Let D,γi,βi>0(i=1,2), Ω⊂R2 be a smooth bounded domain and the hypotheses (H1)–(H5) hold. Then System (1.8) has a unique global nonnegative classical solution (u,v,w)∈(C(¯Ω×[0,∞))∩C2,1(¯Ω×(0,∞)))3, which is uniformly bounded in time, i.e., there exists a constant C>0 independent of t such that
‖u(⋅,t)‖L∞(Ω)+‖v(⋅,t)‖L∞(Ω)+‖w(⋅,t)‖W1,∞(Ω)≤Cforallt>0. | (1.9) |
In particular, one has 0≤w(x,t)≤K0 for all (x,t)∈Ω×(0,∞), where
K0:=max{||w0||L∞(Ω),K}. | (1.10) |
Remark 1.1. For the special case F1(w)=F2(w)=w, h1(u)=u,h2(v)=v and f(w)=μw(1−w), the results of Theorem 1.1 have been obtained in [45]. However, since α1 and α2 may be equal to zero in the hypotheses of this paper, the L2-spatiotemporal estimates of u and v cannot be directly obtained by using the method in [45]. By means of the mechanism "density-suppressed motility", we invoke some ideas used in [14] and apply the self-adjoint realization of Δ+δ with some δ>0 in L2(Ω) to establish the key L2-spatiotemporal estimates of u and v.
The second main aim of this paper is to study the role of non-random dispersal and competition between two predators in the asymptotic properties of the nonnegative spatial homogeneous equilibria of System (1.8). For simplicity, we let F1(w)=F2(w)=w, h1(u)=θ1+α1u, h2(v)=θ2+α2v and f(w)=μw(1−w), where θ1=θ2=θ>0 and α1,α2,μ>0; then, System system (1.8) can be simplified as
{ut=Δ(d1(w)u)+γ1uw−u(θ+α1u)−β1uv,(x,t)∈Ω×(0,∞),vt=Δ(d2(w)v)+γ2vw−v(θ+α2v)−β2uv,(x,t)∈Ω×(0,∞),wt=DΔw−(u+v)w+μw(1−w),(x,t)∈Ω×(0,∞),∂u∂ν=∂v∂ν=∂w∂ν=0,(x,t)∈∂Ω×(0,∞),(u,v,w)(x,0)=(u0,v0,w0)(x),x∈Ω. | (1.11) |
Theorem 1.1 ensures that System (1.11) possesses a unique global bounded nonnegative classical solution (u,v,w) such that 0≤w(x,t)≤K0:=max{||w0||L∞(Ω),1} for all (x,t)∈Ω×(0,∞). Now we find some sufficient conditions of parameters so that (1.11) admits a positive constant solution (u∗,v∗,w∗) satisfying
{γ1w∗−θ−α1u∗−β1v∗=0,γ2w∗−θ−α2v∗−β2u∗=0,−u∗−v∗+μ−μw∗=0, | (1.12) |
i.e.,
AX=B, | (1.13) |
where
A=[−α1−β1γ1−β2−α2γ211μ],X=[u∗v∗w∗],B=[θθμ]. |
If the determinant Φ of the coefficient matrix A in (1.13) does not equal zero, it follows from Cramer's rule that
u∗=ΦuΦ,v∗=ΦvΦ,w∗=ΦwΦ, | (1.14) |
where
{Φ=(γ1+β1μ)(α2−β2)+(γ2+α2μ)(α1−β1),Φu=α2μ(γ1−θ)−β1μ(γ2−θ)+θ(γ1−γ2),Φv=α1μ(γ2−θ)−β2μ(γ1−θ)+θ(γ2−γ1),Φw=(θ+β1μ)(α2−β2)+(θ+α2μ)(α1−β1). | (1.15) |
When β1<α1 and β2<α2, it follows that Φ>0 and Φw>0, and thus we know w∗>0. Next, we shall discuss the sign of Φu and Φv. For convenience, we let
β∗1:=l1μ(γ2−θ) | (1.16) |
and
β∗2:=l2μ(γ1−θ), | (1.17) |
where
l1:=α2μ(γ1−θ)+θ(γ1−γ2) | (1.18) |
and
l2:=α1μ(γ2−θ)+θ(γ2−γ1). | (1.19) |
It is not difficult to see that u∗=ΦuΦ, v∗=ΦvΦ and w∗=ΦwΦ are positive, if γi>θ, i=1,2 and one of the following conditions holds:
(H6) γ1<γ2, l1>0, β1<min{α1,β∗1} and β2<min{α2,β∗2};
(H7) γ1>γ2, l2>0, β1<min{α1,β∗1} and β2<min{α2,β∗2};
(H8) γ1=γ2 and max{β1,β2}<min{α1,α2}.
Now, we give our main results on the asymptotic stability properties of the nonnegative spatial homogeneous equilibria of System (1.11) as follows.
Theorem 1.2. Let Ω⊂R2 be a smooth bounded domain and the parameters γi,αi,βi, (i=1,2), θ, μ and D be positive. Assume that d1(w) and d2(w) satisfy (H1), and that (u,v,w) is a global bounded classical solution of System (1.11). Suppose that γi>θ,i=1,2,
(β1γ2+β2γ1)2<4γ1γ2α1α2 | (1.20) |
and
D>maxw∈[0,K0]w24w∗[u∗|d′1(w)|2γ1d1(w)+v∗|d′2(w)|2γ2d2(w)] | (1.21) |
as well as one of the conditions (H6)–(H8) holds. Then for all initial data (u0,v0,w0) satisfying (H5), there exist positive constants C and λ such that
||u(⋅,t)−u∗||L∞(Ω)+||v(⋅,t)−v∗||L∞(Ω)+||w(⋅,t)−w∗||L∞(Ω)≤Ce−λt | (1.22) |
for all t>0, where (u∗,v∗,w∗) is given by (1.14).
Remark 1.2. From a biological point of view, it is well known that the change of the predators comes from predation, competition and mortality in System (1.11). The parameters γi, βi, αi(i=1,2) and θ respectively stand for the predation rate, competition strength, density-dependent death and natural death rate of the predators, which play a collective role in studying the dynamical behavior of (1.11). More specifically, when γi>θ, it is called strong predation; otherwise, it is weak predation. Hence the results of Theorem 1.2 can tell us that if the predations of two predators are strong and the prey diffusion coefficient D is suitably large, all species can reach a coexistence state.
Theorem 1.3. Let Ω⊂R2 be a smooth bounded domain and the parameters γi,αi,βi, (i=1,2), θ, μ and D be positive. Assume that d1(w) and d2(w) satisfy (H1), and the (u,v,w) is a global bounded classical solution of System (1.11). Suppose that we have (1.20) and
D>maxw∈[0,K0]¯uw2|d′1(w)|24γ1¯wd1(w) | (1.23) |
as well as one of the following conditions holds:
(i) γ1>γ2>θ, l2≤0, β1<min{α1,β∗1} and β2<α2;
(ii) γ1>γ2>θ, l2>0, β1<min{α1,β∗1} and β2∈[β∗2,α2);
(iii) γ1>θ≥γ2,
where
¯u=μ(γ1−θ)α1μ+γ1and¯w=α1μ+θα1μ+γ1. | (1.24) |
Then for all initial data (u0,v0,w0) satisfying (H5), one has
||u(⋅,t)−¯u||L∞(Ω)+||v(⋅,t)||L∞(Ω)+||w(⋅,t)−¯w||L∞(Ω)→0ast→∞, | (1.25) |
exponentially if θ>γ2¯w−β2¯u or algebraically if θ=γ2¯w−β2¯u.
Theorem 1.4. Let Ω⊂R2 be a smooth bounded domain and the parameters γi,αi,βi, (i=1,2), θ, μ and D be positive. Assume that d1(w) and d2(w) satisfy (H1), and the (u,v,w) is a global bounded solution of System (1.11). Suppose that we have (1.20) and
D>maxw∈[0,K0]˜vw2|d′2(w)|24γ2˜wd2(w) | (1.26) |
as well as one of the following conditions holds:
(i) γ2>γ1>θ, l1≤0, β2<min{α2,β∗2} and β1<α1;
(ii) γ2>γ1>θ, l1>0, β2<min{α2,β∗2} and β1∈[β∗1,α1);
(iii) γ2>θ≥γ1,
where
˜v=μ(γ2−θ)α2μ+γ2and˜w=α2μ+θα2μ+γ2. | (1.27) |
Then for all initial data (u0,v0,w0) satisfying (H5), one has
||u(⋅,t)||L∞(Ω)+||v(⋅,t)−˜v||L∞(Ω)+||w(⋅,t)−˜w||L∞(Ω)→0ast→∞, | (1.28) |
exponentially if θ>γ1˜w−β1˜v or algebraically if θ=γ1˜w−β1˜v.
Remark 1.3. It follows from Theorem 1.3 that if the predator u is superior over v in the competition and the prey diffusion coefficient D is suitably large, the semi-trivial equilibrium (¯u,0,¯w) is globally asymptotically stable. Similarly, we can obtain Theorem 1.4. Hence, we only give the conclusion of Theorem 1.4, without showing the details of the proof for brevity.
Theorem 1.5. Let Ω⊂R2 be a smooth bounded domain and the parameters γi,αi,βi, (i=1,2), θ, μ and D be positive. Assume that d1(w) and d2(w) satisfy (H1), and the (u,v,w) is a global bounded solution of System (1.11). Suppose that
γi≤θ,i=1,2. |
Then for all initial data (u0,v0,w0) satisfying (H5), one has
||u(⋅,t)||L∞(Ω)+||v(⋅,t)||L∞(Ω)+||w(⋅,t)−1||L∞(Ω)→0ast→∞, | (1.29) |
exponentially if γi<θ, i=1,2 or algebraically if γi=θ, i=1,2.
Remark 1.4. It follows from Theorem 1.5 that if the capture rates of the two predators are low (i.e. γi≤θ, i=1,2), the prey-only steady state (0,0,1) is globally asymptotically stable regardless of the size of the prey diffusion coefficient D.
Remark 1.5. Compared with the previous results of [33] without competitive terms, the results of Theorems 1.2–1.5 indicate that the competition terms play a crucial role in the global stability of the constant steady states in (1.11). Moreover, under the condition of density-suppressed motility, our global stability results of Theorems 1.2–1.5 can also generalize the ranges of parameters αi and βi(i=1,2) for two dimensional cases in [32]. However, since the heat-semigroup estimates of u and v are no longer applicable due to the appearance of density-suppressed motility, the global stability in L∞-norm is still open in the higher-dimensional problem.
The rest of this paper is organized as follows. In Section 2, we first state the local existence of the classical solution to (1.8) and collect preliminary lemmas. In Section 3, we derive the global existence and boundedness of classical solutions for (1.8) and prove Theorem 1.1. Finally, we shall study the asymptotic stability of global bounded solutions for (1.11) and prove Theorems 1.2–1.5.
In this section, we shall give the local existence and some preliminary lemmas. Firstly, we state the local existence of the classical solution to (1.8), as obtained by means of the abstract theory of quasilinear parabolic systems in [52].
Lemma 2.1. Let D,γi,βi>0(i=1,2), Ω⊂R2 be a smooth bounded domain and the hypotheses (H1)–(H5) hold. Then, there exists a Tmax∈(0,∞] such that System (1.8) possesses a unique classical solution (u,v,w)∈(C(¯Ω×[0,Tmax))∩C2,1(¯Ω×(0,Tmax)))3 satisfying
u,v≥0and0≤w≤K0:=max{||w0||L∞(Ω),K}. | (2.1) |
In addition, the following extensibility criterion holds, i.e. if Tmax<∞, then
lim supt↗Tmax(‖u(⋅,t)‖L∞(Ω)+‖v(⋅,t)‖L∞(Ω)+‖w(⋅,t)‖W1,∞(Ω))=∞. | (2.2) |
Proof. Let z=(u,v,w)T; then, System (1.8) can be rewritten as
{zt=∇⋅(P(z)∇z)+Q(z),(x,t)∈Ω×(0,∞),∂z∂ν=0,(x,t)∈∂Ω×(0,∞),z(⋅,0)=z0=(u0,v0,w0),x∈Ω, | (2.3) |
where
P(z)=(d1(w)0ud′1(w)0d2(w)vd′2(w)00D)andQ(z)=(γ1uF1(w)−uh1(u)−β1uvγ2vF2(w)−vh2(v)−β2uv−uF1(w)−vF2(w)+f(w)). | (2.4) |
According to the conditions that D>0 and di(w)>0(i=1,2), the matrix P(z) is positively definite for the given initial data z0, which asserts that System (1.8) is normally parabolic. Thus it follows from Theorem 7.3 of [53] that there exists a Tmax∈(0,∞] such that System (1.8) admits a unique classical solution (u,v,w)∈(C0(¯Ω×[0,Tmax))∩C2,1(¯Ω×(0,Tmax)))3. The nonnegativity of (u,v,w) directly comes from the maximum principle [14,45]. It similarly follows from Lemma 2.2 of [13] that w≤K0:=max{||w0||L∞(Ω),K}. Since P(z) is an upper triangular matrix, we can deduce from Theorem 5.2 of [54] that the extensibility criterion given by (2.2) holds. The proof of Lemma 2.1 is complete.
Lemma 2.2. Let Ω⊂Rn(n≥1) be a smooth bounded domain, D>0 and T∈(0,∞]. Assume that φ(x,t)∈C(¯Ω×[0,T))∩C2,1(¯Ω×(0,T)) satisfies
{φt=DΔφ−φ+ψ,(x,t)∈Ω×(0,T),∂φ∂ν=0,(x,t)∈∂Ω×(0,T), | (2.5) |
where ψ∈L∞((0,T);Lp(Ω)) with p≥1. Then there exists a positive constant C such that
||φ(⋅,t)||W1,q(Ω)≤Cforallt∈(0,T), | (2.6) |
where
q∈{[1,npn−p),ifp<n,[1,∞),ifp=n,[1,∞],ifp>n. | (2.7) |
Proof. This lemma directly comes from Lemma 1 of [55].
Now, we give the following lemma ([56], Lemma 3.4) to derive some a priori estimates for w.
Lemma 2.3. Let T>0,τ∈(0,T) and a,d>0, and assume that y:[0,T)→[0,∞) is absolutely continuous. If there exists a nonnegative function h∈L1loc([0,T)) satisfying
∫t+τth(s)ds≤dforallt∈[0,T−τ) | (2.8) |
and
y′(t)+ay(t)≤h(t), | (2.9) |
one has
y(t)≤max{y(0)+d,daτ+2d}forallt∈[0,T). | (2.10) |
Next, we give a basic property of the solution components u and v for System (1.8).
Lemma 2.4. Let the assumptions of Lemma 2.1 hold. Then there exists a constant C>0 such that
∫Ωu+vdx≤Cforallt∈(0,Tmax) | (2.11) |
and
∫t+τt∫Ωu2+v2dxds≤Cforallt∈(0,Tmax−τ), | (2.12) |
where τ=min{1,12Tmax}.
Proof. It follows from a direct computation for System (1.8) that
ddt∫Ω(1γ1u+1γ2v+w)dx=∫Ωf(w)dx−1γ1∫Ωuh1(u)dx−β1γ1∫Ωuvdx−1γ2∫Ωvh2(v)dx−β2γ2∫Ωuvdx≤μ∫Ωwdx−1γ1∫Ωu(θ1+α1u)dx−1γ2∫Ωv(θ2+α2v)dx=μ∫Ωwdx−θ1∫Ω1γ1udx−θ2∫Ω1γ2vdx−α1γ1∫Ωu2dx−α2γ2∫Ωv2dx, | (2.13) |
for all t∈(0,Tmax), where we have applied (H3), (H4), β1,β2>0 and (2.1).
Let θ:=min{θ1,θ2}, it follows from (2.1) that
ddt∫Ω(1γ1u+1γ2v+w)dx+α1γ1∫Ωu2dx+α2γ2∫Ωv2dx≤−θ∫Ω(1γ1u+1γ2v+w)dx+(μK0+θ)|Ω|, | (2.14) |
which leads to (2.11) by Gronwall's inequality. If αi>0, i=1,2, then integrating (2.14) over (t,t+τ), we have (2.12) directly. If αi=0, i=1,2, we can also prove (2.12) by means of the idea used in [14]. For the readers' convenience, we give the sketch of the proof.
Let A denote the self-adjoint realization of −Δ+δ under homogeneous Neumann boundary conditions in L2(Ω), where δ∈(0,min{θ1d1(0),θ2d2(0)}). It follows from δ>0 that A has an order-preserving bounded inverse A−1 on L2(Ω). Thus this allows us to obtain a positive constant c1 such that
||A−1Ψ||L2(Ω)≤c1||Ψ||L2(Ω)for allΨ∈L2(Ω) | (2.15) |
and
||A−12Ψ||2L2(Ω)=∫ΩΨ⋅A−1Ψdx≤c1||Ψ||2L2(Ω)for allΨ∈L2(Ω). | (2.16) |
By a simple calculation in (1.8), we have
(1γ1u+1γ2v+w)t=Δ(1γ1d1(w)u+1γ2d2(w)v+Dw)−1γ1uh1(u)−β1γ1uv−1γ2vh2(v)−β2γ2uv+f(w), | (2.17) |
which can be written as
(1γ1u+1γ2v+w)t+A(1γ1d1(w)u+1γ2d2(w)v+Dw)=δ(1γ1d1(w)u+1γ2d2(w)v+Dw)−1γ1uh1(u)−β1γ1uv−1γ2vh2(v)−β2γ2uv+f(w)=1γ1u(δd1(w)−h1(u))+1γ2v(δd2(w)−h2(v))+δDw+f(w)−β1γ1uv−β2γ2uv≤1γ1u(δd1(0)−θ1)+1γ2v(δd2(0)−θ2)+(δD+μ)K0≤(δD+μ)K0:=c2, | (2.18) |
where we have applied (H1), (H3), (H4), (2.1) and δ∈(0,min{θ1d1(0),θ2d2(0)}). Hence, by multiplying (2.18) by A−1(1γ1u+1γ2v+w)≥0 and integrating it over Ω, we derive
12ddt∫Ω|A−12(1γ1u+1γ2v+w)|2dx+∫Ω(1γ1d1(w)u+1γ2d2(w)v+Dw)(1γ1u+1γ2v+w)dx≤c2∫ΩA−1(1γ1u+1γ2v+w)dx. | (2.19) |
According to the fact that 0<di(K0)≤di(w), i=1,2, due to (H1) and (2.1), and by letting c3:=min{d1(K0),d2(K0),D}>0, we deduce
ddt∫Ω|A−12(1γ1u+1γ2v+w)|2dx+2c3∫Ω(1γ1u+1γ2v+w)2dx≤2c2∫ΩA−1(1γ1u+1γ2v+w)dx. | (2.20) |
It follows from Hölder's and Young's inequality as well as (2.15) that
2c2∫ΩA−1(1γ1u+1γ2v+w)dx≤2c2|Ω|12(∫Ω|A−1(1γ1u+1γ2v+w)|2dx)12≤2c1c2|Ω|12(∫Ω(1γ1u+1γ2v+w)2dx)12≤c32∫Ω(1γ1u+1γ2v+w)2dx+2c21c22|Ω|c3. | (2.21) |
According to (2.16), we have
c32c1∫Ω|A−12(1γ1u+1γ2v+w)|2≤c32∫Ω(1γ1u+1γ2v+w)2dx. | (2.22) |
By combining (2.20)–(2.22), we derive
ddt∫Ω|A−12(1γ1u+1γ2v+w)|2dx+c32c1∫Ω|A−12(1γ1u+1γ2v+w)|2+c3∫Ω(1γ1u+1γ2v+w)2dx≤2c21c22|Ω|c3. | (2.23) |
By the ordinary differential equations (ODE) argument, there exists a c4>0 such that
∫Ω|A−12(1γ1u+1γ2v+w)|2dx≤c4, | (2.24) |
which implies that
∫t+τt∫Ω1γ21u2+1γ22v2dxds≤∫t+τt∫Ω(1γ1u+1γ2v)2dxds≤∫t+τt∫Ω(1γ1u+1γ2v+w)2dxds≤c4c3+2c21c22|Ω|c23. | (2.25) |
The proof of Lemma 2.4 is complete.
Finally, we shall give the following key estimate of w, which plays a crucial role in the proof of Theorem 1.1.
Lemma 2.5. Let the assumptions of Lemma 2.1 hold. Then there exists a constant C>0 such that
∫Ω|∇w|2dx≤Cforallt∈(0,Tmax) | (2.26) |
and
∫t+τt∫Ω|Δw|2dxds≤Cforallt∈(0,Tmax−τ), | (2.27) |
where τ=min{1,12Tmax}.
Proof. Multiplying the third equation of System (1.8) with −2Δw and integrating it by parts, we deduce from Young's inequality and (2.1) that
ddt∫Ω|∇w|2dx=−2D∫Ω|Δw|2dx+2∫Ω(uF1(w)+vF2(w))Δwdx−2∫Ωf(w)Δwdx≤−D∫Ω|Δw|2dx+2D∫Ω(uF1(w)+vF2(w))2dx+2D∫Ωf2(w)dx≤−D∫Ω|Δw|2dx+4F21(K0)D∫Ωu2dx+4F22(K0)D∫Ωv2dx+2(μK0)2D|Ω|, | (2.28) |
where we have used the hypotheses (H2) and (H4).
It follows from ∂w∂ν=0, Young's inequality and (2.1) that
∫Ω|∇w|2dx=−∫ΩwΔwdx≤D2∫Ω|Δw|2dx+12D∫Ωw2dx≤D2∫Ω|Δw|2dx+K202D|Ω|. | (2.29) |
Let c5:=max{4F21(K0)D,4F22(K0)D}; we infer from (2.28) and (2.29) that
ddt∫Ω|∇w|2dx+∫Ω|∇w|2dx+D2∫Ω|Δw|2dx≤c5∫Ωu2+v2dx+c6, | (2.30) |
where c6:=2(μK0)2D|Ω|+K202D|Ω|. It follows from Lemma 2.3 and Lemma 2.4 that (2.26) holds. Then integrating (2.30) over (t,t+τ), we can deduce from (2.12) and (2.26) that (2.27) holds.
In this section, we shall study the global existence and uniform boundedness of solutions for system (1.8) when n=2. To do this, we need the following lemmas.
Lemma 3.1. Let the conditions of Theorem 1.1 hold. Then the solution (u,v,w) of system (1.8) satisfies
∫Ωu2dx+∫Ωv2dx≤C | (3.1) |
and
||w(⋅,t)||W1,q(Ω)≤C | (3.2) |
for all q∈[1,∞) and t∈(0,Tmax), where C>0 is a constant independent of t.
Proof. Multiplying the first equation of System (1.8) by 2u and integrating by parts, we deduce from Young's and Hölder's inequalities that
ddt∫Ωu2dx=−2∫Ωd1(w)|∇u|2dx−2∫Ωd′1(w)u∇u⋅∇wdx+2γ1∫Ωu2F1(w)dx−2∫Ωu2h1(u)dx−2β1∫Ωu2vdx≤−∫Ωd1(w)|∇u|2dx+∫Ω|d′1(w)|2d1(w)u2|∇w|2dx+2γ1F1(K0)∫Ωu2dx≤−d1(K0)∫Ω|∇u|2dx+K1∫Ωu2|∇w|2dx+2γ1F1(K0)∫Ωu2dx≤−d1(K0)∫Ω|∇u|2dx+K1(∫Ωu4dx)12(∫Ω|∇w|4dx)12+2γ1F1(K0)∫Ωu2dx, | (3.3) |
where K1:=maxw∈[0,K0]|d′1(w)|2d1(w) and we have applied (H1)–(H3) and (2.1).
By using the Gagliardo-Nirenberg inequality in two dimensions, there exists a C1>0 such that
(∫Ωu4dx)12=||u||2L4(Ω)≤C1(||∇u||L2(Ω)||u||L2(Ω)+||u||2L2(Ω)). | (3.4) |
According to Lemma 2.5 of [19] when n=2, it follows from Lemma 2.5 that
(∫Ω|∇w|4dx)12≤C2(||Δw||L2(Ω)||∇w||L2(Ω)+||∇w||2L2(Ω))≤C3(||Δw||L2(Ω)+1), | (3.5) |
for all t∈(0,Tmax), where C2,C3>0. Thus, by combining (3.4) with (3.5), we infer from Young's inequality that
K1(∫Ωu4dx)12(∫Ω|∇w|4dx)12≤d1(K0)∫Ω|∇u|2dx+C4||u||2L2(Ω)||Δw||2L2(Ω)+C5||u||2L2(Ω), | (3.6) |
where C4,C5 are positive constants. Thus it follows from (3.3) and (3.6) that
ddt∫Ωu2dx≤C6∫Ωu2dx(∫Ω|Δw|2dx+1), | (3.7) |
where C6:=max{C4,C5+2γ1F1(K0)}.
By Lemma 2.4, we can find t0=t0(t)∈((t−τ)+,t) for any t∈(0,Tmax) such that there exists a C7>0 satisfying
∫Ωu2(x,t0)dx≤C7, | (3.8) |
where τ is defined in Lemma 2.4. It follows from Lemma 2.5 that there exists a C8>0 such that
∫t0+τt0∫Ω|Δw(x,t)|2dxdt≤C8. | (3.9) |
Therefore, integrating (3.7) over (t0,t), we deduce from t0<t<t0+τ≤t0+1, (3.8) and (3.9) that
∫Ωu2dx≤∫Ωu2(x,t0)dxeC6∫tt0(∫Ω|Δw|2dx+1)ds≤C7eC6(C8+1) | (3.10) |
for all t∈(0,Tmax).
Similarly, we obtain
∫Ωv2dx≤C9for allt∈(0,Tmax). | (3.11) |
It follows from the third equation of System (1.8), we know that w solves the following problem
{wt=DΔw−w+g(u,v,w),(x,t)∈Ω×(0,Tmax),∂w∂ν=0,(x,t)∈∂Ω×(0,Tmax), | (3.12) |
where g(u,v,w)=w−uF1(w)−vF2(w)+f(w). According to (H2), (H3) and (2.1), we infer from (3.10) and (3.11) that
||g(u,v,w)||L2(Ω)≤C10(||u||L2(Ω)+||v||L2(Ω)+1)≤C11 | (3.13) |
for all t∈(0,Tmax). Hence, it follows from Lemma 2.2 in two dimensions that (3.2) holds. The proof of Lemma 3.1 is complete.
Next, we shall prove the boundedness of w in W1∞(Ω).
Lemma 3.2. Let the conditions of Theorem 1.1 hold. Then the solution component w of system (1.8) satisfies
||w(⋅,t)||W1,∞(Ω)≤C | (3.14) |
for all t∈(0,Tmax), where C>0 is a constant independent of t.
Proof. Multiplying the first equation of System (1.8) by u2 and integrating by parts, we deduce from Young's and Hölder's inequalities that
13ddt∫Ωu3dx=−2∫Ωd1(w)u|∇u|2dx−2∫Ωd′1(w)u2∇u⋅∇wdx+γ1∫Ωu3F1(w)dx−∫Ωu3h1(u)dx−β1∫Ωu3vdx≤−∫Ωd1(w)u|∇u|2dx+∫Ω|d′1(w)|2d1(w)u3|∇w|2dx+γ1F1(K0)∫Ωu3dx−θ1∫Ωu3dx≤−4d1(K0)9∫Ω|∇u32|2dx+K1∫Ωu3|∇w|2dx+γ1F1(K0)∫Ωu3dx−θ1∫Ωu3dx≤−4d1(K0)9∫Ω|∇u32|2dx+K1(∫Ωu6dx)12(∫Ω|∇w|4dx)12+γ1F1(K0)∫Ωu3dx−θ1∫Ωu3dx, | (3.15) |
for all t∈(0,Tmax), where K1 is defined in the proof of Lemma 3.1 and we have applied (H1)–(H3) and (2.1).
It follows from Lemma 3.1 that there exist positive constants C1 and C2 such that ||∇w||L4(Ω)≤C1 and ||u||L2(Ω)≤C2 for all t∈(0,Tmax). Then by using the Gagliardo-Nirenberg inequality and Young's inequality, we can find positive constants Ci, i=3,⋅⋅⋅,6 such that
K1(∫Ωu6dx)12(∫Ω|∇w|4dx)12≤K1C21||u32||2L4(Ω)≤C3(||∇u32||43L2(Ω)⋅||u32||23L43(Ω)+||u32||2L43(Ω))≤2d1(K0)9∫Ω|∇u32|2dx+C4 | (3.16) |
and
γ1F1(K0)∫Ωu3dx≤γ1F1(K0)||u32||2L2(Ω)≤C5(||∇u32||23L2(Ω)⋅||u32||43L43(Ω)+||u32||2L43(Ω))≤2d1(K0)9∫Ω|∇u32|2dx+C6 | (3.17) |
for all t∈(0,Tmax).
Combining (3.15)–(3.17), we have
ddt∫Ωu3dx+3θ1∫Ωu3dx≤C7:=3(C4+C6) | (3.18) |
for all t∈(0,Tmax). By the ODE argument, we can derive
∫Ωu3dx≤max{∫Ωu30dx,C73θ1}for allt∈(0,Tmax). | (3.19) |
Similarly, we also derive the boundedness of ||v||L3(Ω). Then it follows from Lemma 2.2 in two dimensions that (3.14) holds.
By means of the boundedness of ||w(⋅,t)||W1,∞(Ω), it follows from the Moser iteration of [45] that we can obtain the boundedness of ||u(⋅,t)||L∞(Ω) and ||v(⋅,t)||L∞(Ω) for all t∈(0,Tmax).
Lemma 3.3. Let the conditions of Theorem 1.1 hold. Then the solution component (u,v) of system (1.8) satisfies
||u(⋅,t)||L∞(Ω)+||v(⋅,t)||L∞(Ω)≤C | (3.20) |
for all t∈(0,Tmax), where C>0 is a constant independent of t.
Proof. Multiplying the first equation of System (1.8) by up−1 with p≥2 and integrating by parts, we deduce from Young's inequality that
1pddt∫Ωupdx=−(p−1)∫Ωd1(w)up−2|∇u|2dx−(p−1)∫Ωd′1(w)up−1∇u⋅∇wdx+γ1∫ΩupF1(w)dx−∫Ωuph1(u)dx−β1∫Ωupvdx≤−p−12∫Ωd1(w)up−2|∇u|2dx+p−12∫Ω|d′1(w)|2d1(w)up|∇w|2dx+γ1F1(K0)∫Ωupdx≤−p−12d1(K0)∫Ωup−2|∇u|2dx+p−12K1∫Ωup|∇w|2dx+γ1F1(K0)∫Ωupdx | (3.21) |
for all t∈(0,Tmax), where K1 is defined in the proof of Lemma 3.1 and we have applied (H1)–(H3) and (2.1).
It follows from Lemma 3.2 that there exists a C1>0 such that ||∇w(⋅,t)||L∞(Ω)≤C1 for all t∈(0,Tmax). Hence, we deduce from (3.21) that
ddt∫Ωupdx+p(p−1)d1(K0)2∫Ωup−2|∇u|2dx+p(p−1)∫Ωupdx≤C2p(p−1)∫Ωupdx, | (3.22) |
for all t∈(0,Tmax), where C2:=K1C212+γ1F1(K0)+1 is independent of p. The rest can be handled exactly as the Moser iteration in Lemma 2.7 of [45] to derive the boundedness of ||u(⋅,t)||L∞(Ω) for all t∈(0,Tmax). Similarly, we can obtain the boundedness of ||v(⋅,t)||L∞(Ω) for all t∈(0,Tmax). The proof of Lemma 3.3 is complete.
Proof of Theorem 1.1. Theorem 1.1 is a direct consequence of Lemma 2.1, Lemma 3.2 and Lemma 3.3.
In this section, we shall study the asymptotic stability of global bounded solutions for System (1.11) by constructing energy functionals used in [13,57]. To do this, we first give some regularity results of the solution (u,v,w) for System (1.11).
Lemma 4.1. Let (u,v,w) be a global bounded classical solution for (1.11) ensured in Theorem 1.1. Then there exist σ∈(0,1) and C>0 such that
||u||Cσ,σ2(¯Ω×[t,t+1])+||v||Cσ,σ2(¯Ω×[t,t+1])+||w||C2+σ,1+σ2(¯Ω×[t,t+1])≤Cforallt>1. | (4.1) |
Proof. This lemma can be verified by a similar argument in Lemma 4.1 of [14], so we omit the details here for brevity.
Lemma 4.2. Let (u,v,w) be a global bounded classical solution for (1.11) ensured in Theorem 1.1. Then there exists a C>0 such that
||u(⋅,t)||W1,4(Ω)+||v(⋅,t)||W1,4(Ω)≤Cforallt>0. | (4.2) |
Proof. This lemma can be verified by a similar argument in Lemma 3.6 of [45], so we omit the details here for brevity.
In order to prove the asymptotic stabilization of global bounded solutions for system (1.11), we provide the following lemma, which is proved in [57].
Lemma 4.3. Let ϕ:(1,∞)→[0,∞) be uniformly continuous such that ∫∞1ϕ(t)dt<∞. Then
ϕ(t)→0ast→∞. | (4.3) |
In this subsection, we are devoted to studying the stabilization of the coexistence steady state (u∗,v∗,w∗) for some parameters cases. Let us introduce the following functionals
E1(t)=1γ1∫Ω(u−u∗−u∗lnuu∗)dx+1γ2∫Ω(v−v∗−v∗lnvv∗)dx+∫Ω(w−w∗−w∗lnww∗)dx, |
and
F1(t)=∫Ω(u−u∗)2dx+∫Ω(v−v∗)2dx+∫Ω(w−w∗)2dx+∫Ω|∇uu|2dx+∫Ω|∇vv|2dx+∫Ω|∇w|2dx, |
where (u∗,v∗,w∗) is given by (1.14).
Lemma 4.4. Let the conditions of Theorem 1.2 hold. Then there exists a positive constant ε1 independent of t such that
E1(t)≥0 | (4.4) |
and
ddtE1(t)≤−ε1F1(t)forallt>0. | (4.5) |
Proof. Let
I1(t):=1γ1∫Ω(u−u∗−u∗lnuu∗)dx,I2(t):=1γ2∫Ω(v−v∗−v∗lnvv∗)dx,I3(t):=∫Ω(w−w∗−w∗lnww∗)dx, |
then E1(t) can be rewritten as
E1(t)=I1(t)+I2(t)+I3(t)for allt>0. |
Step 1: We shall prove the nonnegativity of E1(t) for all t>0. Let H(ξ):=ξ−u∗lnξ for ξ>0; it follows from Taylor's formula for all x∈Ω and each t>0 that there exists a τ=τ(x,t)∈(0,1) such that
u−u∗−u∗lnuu∗=H(u)−H(u∗)=H′(u∗)⋅(u−u∗)+12H′′(τu+(1−τ)u∗)⋅(u−u∗)2=u∗2(τu+(1−τ)u∗)2(u−u∗)2≥0. |
Hence, we immediately derive that I1(t)=∫Ω(H(u)−H(u∗))dx≥0. Similarly, we know that I2(t)≥0 and I3(t)≥0 for all t>0. Thus, we know that (4.4) holds.
Step 2: Now, we further prove (4.5). By a series of simple calculations, we get
ddtI1(t)=1γ1∫Ωu−u∗uutdx=1γ1∫Ωu−u∗u(Δ(d1(w)u)+γ1uw−u(θ+α1u)−β1uv)dx=−u∗γ1∫Ωd1(w)|∇u|2u2dx−u∗γ1∫Ωd′1(w)∇u⋅∇wudx+1γ1∫Ω(u−u∗)(γ1w−θ−α1u−β1v)dx=−u∗γ1∫Ωd1(w)|∇u|2u2dx−u∗γ1∫Ωd′1(w)∇u⋅∇wudx+∫Ω(u−u∗)(w−w∗)dx−α1γ1∫Ω(u−u∗)2dx−β1γ1∫Ω(u−u∗)(v−v∗)dx, | (4.6) |
where we have used the fact that θ=γ1w∗−α1u∗−β1v∗.
Similarly, it follows from the identities θ=γ2w∗−β2u∗−α2v∗ and μ=u∗+v∗+μw∗ that
ddtI2(t)=−v∗γ2∫Ωd2(w)|∇v|2v2dx−v∗γ2∫Ωd′2(w)∇v⋅∇wvdx+∫Ω(v−v∗)(w−w∗)dx−α2γ2∫Ω(v−v∗)2dx−β2γ2∫Ω(v−v∗)(u−u∗)dx, | (4.7) |
and
ddtI3(t)=−Dw∗∫Ω|∇w|2w2dx−∫Ω(w−w∗)(u−u∗)dx−∫Ω(w−w∗)(v−v∗)dx−μ∫Ω(w−w∗)2dx. | (4.8) |
Hence, by combining (4.6)–(4.8), we derive
ddtE1(t)=−α1γ1∫Ω(u−u∗)2dx−α2γ2∫Ω(v−v∗)2dx−μ∫Ω(w−w∗)2dx−(β1γ1+β2γ2)∫Ω(u−u∗)(v−v∗)dx−u∗γ1∫Ωd1(w)|∇u|2u2dx−v∗γ2∫Ωd2(w)|∇v|2v2dx−Dw∗∫Ω|∇w|2w2dx−u∗γ1∫Ωd′1(w)∇u⋅∇wudx−v∗γ2∫Ωd′2(w)∇v⋅∇wvdx:=−∫ΩX1A1XT1dx−∫ΩY1B1YT1dx, | (4.9) |
where X1=(u−u∗,v−v∗,w−w∗) and Y1=(∇uu,∇vv,∇w), as well as
A1=(α1γ112(β1γ1+β2γ2)012(β1γ1+β2γ2)α2γ2000μ),B1=(u∗d1(w)γ10u∗d′1(w)2γ10v∗d2(w)γ2v∗d′2(w)2γ2u∗d′1(w)2γ1v∗d′2(w)2γ2Dw∗w2). | (4.10) |
It follows from (1.20) that
|α1γ1|>0and|α1γ112(β1γ1+β2γ2)12(β1γ1+β2γ2)α2γ2|=α1α2γ1γ2−14(β1γ1+β2γ2)2>0 | (4.11) |
as well as
|A1|=μ(α1α2γ1γ2−14(β1γ1+β2γ2)2)>0, | (4.12) |
which implies that the matrix A1 is positive definite as according to Sylvester's criterion. Similarly, we deduce from (1.21) that
|u∗d1(w)γ1|>0and|u∗d1(w)γ100v∗d2(w)γ2|=u∗v∗d1(w)d2(w)γ1γ2>0 | (4.13) |
as well as
|B1|=u∗v∗w∗d1(w)d2(w)γ1γ2w2(D−u∗w2|d′1(w)|24γ1w∗d1(w)−v∗w2|d′2(w)|24γ2w∗d2(w))>0, | (4.14) |
which implies that the matrix B1 is positive definite. Thus there exist positive constants κ1 and κ2 such that
X1A1XT1≥κ1|X1|2andY1B1YT1≥κ2|Y1|2 | (4.15) |
for all x∈Ω and t>0. Let ε1:=min{κ1,κ2}, we have
ddtE1(t)≤−ε1∫Ω|X1|2+|Y1|2dxfor allt>0, | (4.16) |
which implies that (4.5) holds. The proof of Lemma 4.4 is complete.
With the aid of Lemma 4.4, we shall give the following large time behavior of global solutions for system (1.11).
Lemma 4.5. Let the assumptions of Theorem 1.2 hold. Then the global bounded solution of (1.11) converges to the coexistence steady state (u∗,v∗,w∗) given by (1.14), i.e.,
||u(⋅,t)−u∗||L∞(Ω)+||v(⋅,t)−v∗||L∞(Ω)+||w(⋅,t)−w∗||L∞(Ω)→0 | (4.17) |
as t→∞.
Proof. It follows from Lemma 4.4 and integration over (1,∞) that
∫∞1F1(t)dt≤E1(1)ε1<∞. |
According to Theorem 1.1 and Lemma 4.1, the bounded solution u,v and w are Hölder continuous in ˉΩ×[t,t+1] with respect to t>1. Thus we conclude that F1(t) is uniformly continuous in (1,∞). Thus we infer from Lemma 4.3 that
∫Ω(u−u∗)2dx+∫Ω(v−v∗)2dx+∫Ω(w−w∗)2dx→0 | (4.18) |
as t→∞. By the Gagliardo-Nirenberg inequality in two dimensions, there exists a C1>0 such that
‖u−u∗‖L∞(Ω)≤C1‖u−u∗‖23W1,4(Ω)‖u−u∗‖13L2(Ω). |
Moreover, it follows from Lemma 4.2 that u(⋅,t)−u∗ is bounded in W1,4(Ω); thus, we conclude from (4.18) that u(⋅,t)→u∗ in L∞(Ω) as t→∞. By the similar arguments for v and w, we derive (4.17). The proof of Lemma 4.5 is complete.
Now, we give the convergence rate of the coexistence state (u∗,v∗,w∗) for System (1.11).
Lemma 4.6. Let the assumptions of Theorem 1.2 hold; the global bounded solution (u,v,w) of (1.11) exponentially converges to the coexistence state (u∗,v∗,w∗), i.e. there exist C>0 and λ>0 such that
||u(⋅,t)−u∗||L∞(Ω)+||v(⋅,t)−v∗||L∞(Ω)+||w(⋅,t)−w∗||L∞(Ω)≤Ce−λt | (4.19) |
for all t>T1, where T1>0 is some fixed time.
Proof. It follows from Lemma 4.5 that ||u−u∗||L∞(Ω)→0 as t→∞. Therefore, we apply L'Hôpital's rule to obtain
limu→u∗u−u∗−u∗lnuu∗(u−u∗)2=12u∗, | (4.20) |
which implies that there exists a t1>0 such that
14u∗∫Ω(u−u∗)2dx≤∫Ω(u−u∗−u∗lnuu∗)dx≤34u∗∫Ω(u−u∗)2dx | (4.21) |
for all t>t1. Similarly, we can find t2>0 satisfying
14v∗∫Ω(v−v∗)2dx≤∫Ω(v−v∗−v∗lnvv∗)dx≤34v∗∫Ω(v−v∗)2dx | (4.22) |
and
14w∗∫Ω(w−w∗)2dx≤∫Ω(w−w∗−w∗lnww∗)dx≤34w∗∫Ω(w−w∗)2dx | (4.23) |
for all t>t2. Let T1:=max{t1,t2}; by means of the definitions of E1(t) and F1(t), it follows from the second inequalities in (4.21)–(4.23) that there exists a C1>0 such that
C1E1(t)≤F1(t)for allt>T1. | (4.24) |
By Lemma 4.4, we derive
E′1(t)≤−ε1F1(t)≤−ε1C1E1(t)for allt>T1, | (4.25) |
which implies that there exist C2>0 and C3>0 such that
E1(t)≤C2e−C3(t−T1)for allt>T1. | (4.26) |
Thus we deduce from the first inequalities in (4.21)–(4.23) that there exists a C4>0 such that
∫Ω(u(x,t)−u∗)2dx+∫Ω(v(x,t)−v∗)2dx+∫Ω(w(x,t)−w∗)2dx≤C4E1(t)≤C2C4e−C3(t−T1)for allt>T1. | (4.27) |
It follows from the Gagliardo-Nirenberg inequality in two dimensions, Lemma 4.2 and Lemma 3.2 that there exist positive constants C5 and C6 such that
||u−u∗||L∞(Ω)+||v−v∗||L∞(Ω)+||w−w∗||L∞(Ω)≤C5(||u−u∗||23W1,4(Ω)||u−u∗||13L2(Ω)+||v−v∗||23W1,4(Ω)||v−v∗||13L2(Ω)+||w−w∗||23W1,4(Ω)||w−w∗||13L2(Ω))≤C6(∫Ω(u−u∗)2dx+∫Ω(v−v∗)2dx+∫Ω(w−w∗)2dx)16≤C6(C2C4)16e−C3(t−T1)6 | (4.28) |
for all t>T1. The proof of Lemma 4.6 is complete.
Proof of Theorem 1.2. The statement of Theorem 1.2 is a straightforward consequence of Lemma 4.6.
In this subsection, we shall study the stabilization of the semi-trivial steady state (¯u,0,¯w) or (0,˜v,˜w) for some parameters cases. Since the methods of the proofs of Theorem 1.3 and Theorem 1.4 are very similar, we only give the proof of Theorem 1.3 for brevity. To do this, let us introduce the following functionals
E2(t)=1γ1∫Ω(u−¯u−¯ulnu¯u)dx+1γ2∫Ωvdx+∫Ω(w−¯w−¯wlnw¯w)dx, |
and
F2(t)=∫Ω(u−¯u)2dx+∫Ωv2dx+∫Ω(w−¯w)2dx+∫Ω|∇uu|2dx+∫Ω|∇w|2dx, |
where ¯u=μ(γ1−θ)α1μ+γ1and¯w=α1μ+θα1μ+γ1.
Lemma 4.7. Let the conditions of Theorem 1.3 hold. Then there exists a positive constant ε2 independent of t such that
E2(t)≥0 | (4.29) |
and
ddtE2(t)≤−ε2F2(t)−1γ2(θ−γ2¯w+β2¯u)∫Ωvdxforallt>0. | (4.30) |
Proof. Let
J1(t):=1γ1∫Ω(u−¯u−¯ulnu¯u)dx,J2(t):=1γ2∫Ωvdx,J3(t):=∫Ω(w−¯w−¯wlnw¯w)dx, |
then E2(t) can be represented as
E2(t)=J1(t)+J2(t)+J3(t)for allt>0. |
Firstly, we can prove the nonnegativity of E2(t) for all t>0 by the similar arguments used in Step 1 in Lemma 4.4. For brevity, we omit the details here. Now, we shall prove (4.30). By a series of simple calculations, we get
ddtJ1(t)=−¯uγ1∫Ωd1(w)|∇u|2u2dx−¯uγ1∫Ωd′1(w)∇u⋅∇wudx+∫Ω(u−¯u)(w−¯w)dx−α1γ1∫Ω(u−¯u)2dx−β1γ1∫Ω(u−¯u)vdx, | (4.31) |
where we have used the fact that θ=γ1¯w−α1¯u.
Similarly, we can derive
ddtJ2(t)=−α2γ2∫Ωv2dx−β2γ2∫Ωv(u−¯u)dx+∫Ωv(w−¯w)dx−1γ2(θ−γ2¯w+β2¯u)∫Ωvdx, | (4.32) |
and
ddtJ3(t)=−D¯w∫Ω|∇w|2w2dx−∫Ω(w−¯w)(u−¯u)dx−∫Ω(w−¯w)vdx−μ∫Ω(w−¯w)2dx, | (4.33) |
where we have used the fact that μ=¯u+μ¯w. Thus it follows from (4.31)–(4.33) that
ddtE2(t)=−α1γ1∫Ω(u−¯u)2dx−α2γ2∫Ωv2dx−μ∫Ω(w−¯w)2dx−(β1γ1+β2γ2)∫Ω(u−¯u)vdx−¯uγ1∫Ωd1(w)|∇u|2u2dx−D¯w∫Ω|∇w|2w2dx−¯uγ1∫Ωd′1(w)∇u⋅∇wudx−1γ2(θ−γ2¯w+β2¯u)∫Ωvdx:=−∫ΩX2A2XT2dx−∫ΩY2B2YT2dx−1γ2(θ−γ2¯w+β2¯u)∫Ωvdx, | (4.34) |
where X2=(u−¯u,v,w−¯w) and Y2=(∇uu,∇w), as well as
A2=(α1γ112(β1γ1+β2γ2)012(β1γ1+β2γ2)α2γ2000μ),B2=(¯ud1(w)γ1¯ud′1(w)2γ1¯ud′1(w)2γ1D¯ww2). | (4.35) |
It follows from (1.20) that
|α1γ1|>0and|α1γ112(β1γ1+β2γ2)12(β1γ1+β2γ2)α2γ2|=α1α2γ1γ2−14(β1γ1+β2γ2)2>0 | (4.36) |
as well as
|A2|=μ(α1α2γ1γ2−14(β1γ1+β2γ2)2)>0, | (4.37) |
which implies that the matrix A2 is positive definite as according to Sylvester's criterion. Similarly, we deduce from (1.23) that
|¯ud1(w)γ1|>0and|B2|=¯u¯wd1(w)γ1w2(D−¯uw2|d′1(w)|24γ1¯wd1(w))>0, | (4.38) |
which implies that the matrix B2 is positive definite. Thus there exist positive constants ι1 and ι2 such that
X2A2XT2≥ι1|X2|2andY2B2YT2≥ι2|Y2|2 | (4.39) |
for all x∈Ω and t>0. Let ε2∈(0,min{ι1,ι2}); we obtain
ddtE2(t)≤−ε2∫Ω|X2|2+|Y2|2dx−1γ2(θ−γ2¯w+β2¯u)∫Ωvdxfor allt>0, | (4.40) |
which implies that (4.30) holds. The proof of Lemma 4.7 is complete.
With the help of Lemma 4.7, we shall give the following stabilization of the semi-trivial steady state (¯u,0,¯w) for System (1.11).
Lemma 4.8. Let the assumptions of Theorem 1.3 hold. Then the global bounded solution (u,v,w) of (1.11) converges to the semi-trivial steady state (¯u,0,¯w) given by (1.24), i.e.,
||u(⋅,t)−¯u||L∞(Ω)+||v(⋅,t)||L∞(Ω)+||w(⋅,t)−¯w||L∞(Ω)→0 | (4.41) |
as t→∞.
Proof. The proof of this lemma is similar to that of Lemma 4.5; here we omit the details.
Now, we give the convergence rate of the semi-trivial steady state (¯u,0,¯w) for System (1.11).
Lemma 4.9. Let the assumptions of Theorem 1.3 hold; then, there exist positive constants C and λ such that:
(a) when θ=γ2¯w−β2¯u, then
||u(⋅,t)−¯u||L∞(Ω)+||v(⋅,t)||L∞(Ω)+||w(⋅,t)−¯w||L∞(Ω)≤C(1+t)−λfor allt>T2; | (4.42) |
(b) when θ>γ2¯w−β2¯u, then
||u(⋅,t)−¯u||L∞(Ω)+||v(⋅,t)||L∞(Ω)+||w(⋅,t)−¯w||L∞(Ω)≤Ce−λtfor allt>T2, | (4.43) |
where T2>0 is some fixed time.
Proof. Let
F∗2(t):=∫Ω(u−¯u)2dx+∫Ωv2dx+∫Ω(w−¯w)2dx, | (4.44) |
then it follows from Lemma 4.7 that there exists a ε2>0 such that
ddtE2(t)≤−ε2F∗2(t)−1γ2(θ−γ2¯w+β2¯u)∫Ωvdxfor all t > 0 . | (4.45) |
We deduce from Lemma 4.8 that ||u(⋅,t)−¯u||L∞(Ω)+||v(⋅,t)||L∞(Ω)+||w(⋅,t)−¯w||L∞(Ω)→0 as t→∞. Hence, we apply L'Hôpital's rule to obtain
limu→¯uu−¯u−¯ulnu¯u(u−¯u)2=12¯u, | (4.46) |
which implies that there exists a t′1>0 such that
14¯u∫Ω(u−¯u)2dx≤∫Ω(u−¯u−¯ulnu¯u)dx≤34¯u∫Ω(u−¯u)2dx | (4.47) |
for all t>t′1. Similarly, we can find t′2>0 satisfying
14¯w∫Ω(w−¯w)2dx≤∫Ω(w−¯w−¯wlnw¯w)dx≤34¯w∫Ω(w−¯w)2dx | (4.48) |
for all t>t′2.
By using the fact that lims→0ss2+s=1, it follows from ||v(⋅,t)||L∞(Ω)→0 as t→∞ that there exists a t′3>0 such that
12∫Ωv2+vdx≤∫Ωvdx≤32∫Ωv2+vdx | (4.49) |
for all t>t′3.
(a) When θ=γ2¯w−β2¯u, (4.45) can be turned into
ddtE2(t)≤−ε2F∗2(t)for all t > 0 . | (4.50) |
Let T2:=max{t′1,t′2,t′3}; by means of the definitions of E2(t) and F∗2(t), it follows from the second inequalities in (4.47) and (4.48) that there exist positive constants C1 and C2 such that
E2(t)≤34γ1¯u∫Ω(u−¯u)2dx+1γ2∫Ωvdx+34¯w∫Ω(w−¯w)2dx≤C1(∫Ω(u−¯u)2dx)12+C1(∫Ωv2dx)12+C1(∫Ω(w−¯w)2dx)12≤C2(F∗2(t))12, | (4.51) |
for all t>T2, where we have used Hölder's inequality and the boundedness of (u,v,w) asserted by Theorem 1.1. Thus, we deduce from (4.50) that
E′2(t)≤−ε2C22E22(t)for allt>T2, | (4.52) |
which implies
E2(t)≤C3t−T2for allt>T2, | (4.53) |
with some positive constant C3. Hence we infer from the first inequalities in (4.47)–(4.49) that there exists a C4>0 such that
∫Ω(u−¯u)2dx+∫Ωv2dx+∫Ω(w−¯w)2dx≤C4E2(t)≤C3C4t−T2for allt>T2. | (4.54) |
It follows from the Gagliardo-Nirenberg inequality in two dimensions, Lemma 4.2 and Lemma 3.2 that there exist positive constants C5 and C6 such that
||u−¯u||L∞(Ω)+||v||L∞(Ω)+||w−¯w||L∞(Ω)≤C5(||u−¯u||23W1,4(Ω)||u−¯u||13L2(Ω)+||v||23W1,4(Ω)||v||13L2(Ω)+||w−¯w||23W1,4(Ω)||w−¯w||13L2(Ω))≤C6(∫Ω(u−¯u)2dx+∫Ωv2dx+∫Ω(w−¯w)2dx)16≤C6(C3C4)16(t−T2)−16 | (4.55) |
for all t>T2.
(b) When θ>γ2¯w−β2¯u, let T2:=max{t′1,t′2,t′3}; by means of the definitions of E2(t) and F∗2(t), it follows from the second inequalities in (4.47) and (4.48) that there exists a positive constant C7 such that
E2(t)≤C7(F∗2(t)+∫Ωvdx), | (4.56) |
for all t>T2.
By combining (4.45) with (4.56), we have
ddtE2(t)≤−ε2C7E2(t)−1γ2(θ−γ2¯w+β2¯u−γ2ε2)∫Ωvdxfor allt>T2. | (4.57) |
Since θ>γ2¯w−β2¯u, then we can select ε2≤1γ2(θ−γ2¯w+β2¯u) such that
ddtE2(t)≤−ε2C7E2(t)for allt>T2, | (4.58) |
which means that there exist C8>0 and C9>0 satisfying
E2(t)≤C8e−C9(t−T2)for allt>T2. | (4.59) |
Thus we deduce from the first inequalities in (4.47)–(4.49) that there exists a C10>0 such that
∫Ω(u−¯u)2dx+∫Ωv2dx+∫Ω(w−¯w)2dx≤C10E2(t)≤C8C10e−C9(t−T2)for allt>T2. | (4.60) |
It follows from the Gagliardo-Nirenberg inequality in two dimensions, Lemma 4.2 and Lemma 3.2 that there exist positive constants C11 and C12 such that
||u−¯u||L∞(Ω)+||v||L∞(Ω)+||w−¯w||L∞(Ω)≤C11(||u−¯u||23W1,4(Ω)||u−¯u||13L2(Ω)+||v||23W1,4(Ω)||v||13L2(Ω)+||w−¯w||23W1,4(Ω)||w−¯w||13L2(Ω))≤C12(∫Ω(u−¯u)2dx+∫Ωv2dx+∫Ω(w−¯w)2dx)16≤C12(C8C10)16e−C9(t−T2)6 | (4.61) |
for all t>T2. The proof of Lemma 4.9 is complete.
Proof of Theorem 1.3. The statement of Theorem 1.3 is a direct consequence of Lemma 4.9.
In this subsection, we are devoted to discussing the asymptotic stability of the prey-only steady state (0,0,1) under some suitable parameters conditions. To do this, let us denote the following functionals
E3(t)=1γ1∫Ωudx+1γ2∫Ωvdx+∫Ω(w−1−lnw)dx |
and
F3(t)=∫Ωu2dx+∫Ωv2dx+∫Ω(w−1)2dx+∫Ω|∇w|2dx, |
we can derive the following estimates of E3(t) and F3(t).
Lemma 4.10. Let the conditions of Theorem 1.5 hold. Then there exists a ε3>0 independent of t such that
E3(t)≥0 | (4.62) |
and
ddtE3(t)≤−ε3F3(t)−1γ1(θ−γ1)∫Ωudx−1γ2(θ−γ2)∫Ωvdxforallt>0. | (4.63) |
Proof. By the similar arguments as in the proofs of Lemma 4.4 and Lemma 4.7, we can derive (4.62) and
ddtE3(t)=−α1γ1∫Ωu2dx−α2γ2∫Ωv2dx−μ∫Ω(w−1)2dx−(β1γ1+β2γ2)∫Ωuvdx−D∫Ω|∇w|2w2dx−1γ1(θ−γ1)∫Ωudx−1γ2(θ−γ2)∫Ωvdx≤−α1γ1∫Ωu2dx−α2γ2∫Ωv2dx−μ∫Ω(w−1)2dx−DK20∫Ω|∇w|2dx−1γ1(θ−γ1)∫Ωudx−1γ2(θ−γ2)∫Ωvdx, | (4.64) |
where we have used the fact that w≤K0=max{||w0||L∞(Ω),1}. Let ε3∈(0,min{α1γ1,α2γ2,μ,DK20}); we obtain
ddtE3(t)≤−ε3F3(t)−1γ1(θ−γ1)∫Ωudx−1γ2(θ−γ2)∫Ωvdx, | (4.65) |
for all t>0. The proof of Lemma 4.10 is complete.
With the help of Lemma 4.10, we shall give the following stabilization of the prey-only steady state for System (1.11).
Lemma 4.11. Let the assumptions of Theorem 1.5 hold. Then the global bounded solution of (1.11) converges to the prey-only steady state (0,0,1), i.e.,
||u(⋅,t)||L∞(Ω)+||v(⋅,t)||L∞(Ω)+||w(⋅,t)−1||L∞(Ω)→0 | (4.66) |
as t→∞.
Proof. The proof of this lemma is similar to that of Lemma 4.5; here we omit the details.
Now, we give the convergence rate of the prey-only steady state (0,0,1) for System (1.11).
Lemma 4.12. Let the assumptions of Theorem 1.5 hold; then there exist positive constants C and λ such that:
(a) when γi=θ, i=1,2, then
||u(⋅,t)||L∞(Ω)+||v(⋅,t)||L∞(Ω)+||w(⋅,t)−1||L∞(Ω)≤C(1+t)−λforallt>T3; | (4.67) |
(b) when γi<θ, i=1,2, then
||u(⋅,t)||L∞(Ω)+||v(⋅,t)||L∞(Ω)+||w(⋅,t)−1||L∞(Ω)≤Ce−λtforallt>T3, | (4.68) |
where T3>0 is some fixed time.
Proof. Let
F∗3(t):=∫Ωu2dx+∫Ωv2dx+∫Ω(w−1)2dx, | (4.69) |
then it follows from Lemma 4.10 that there exists a ε3>0 such that
ddtE3(t)≤−ε3F∗3(t)−1γ1(θ−γ1)∫Ωudx−1γ2(θ−γ2)∫Ωvdxfor all t > 0 . | (4.70) |
By using the facts that lims→0ss2+s=1 and lims→1s−1−lns(s−1)2=12, it follows from ||u(⋅,t)||L∞(Ω)+||v(⋅,t)||L∞(Ω)+||w(⋅,t)−1||L∞(Ω)→0 as t→∞, as asserted in Lemma 4.11 that there exists a T3>0 such that
12∫Ωu2+udx≤∫Ωudx≤32∫Ωu2+udx | (4.71) |
and
12∫Ωv2+vdx≤∫Ωvdx≤32∫Ωv2+vdx | (4.72) |
as well as
14∫Ω(w−1)2dx≤∫Ω(w−1−lnw)dx≤34∫Ω(w−1)2dx | (4.73) |
for all t>T3.
(a) When γi=θ, i=1,2, (4.70) can be simplified as
ddtE3(t)≤−ε3F∗3(t)for all t > 0 . | (4.74) |
By means of the definitions of E3(t) and F∗3(t), it follows from the second inequality in (4.73) that there exist positive constants C1 and C2 such that
E3(t)≤1γ1∫Ωudx+1γ2∫Ωvdx+34∫Ω(w−1)2dx≤C1(∫Ωu2dx)12+C1(∫Ωv2dx)12+C1(∫Ω(w−1)2dx)12≤C2(F∗3(t))12, | (4.75) |
for all t>T3, where we have used Hölder's inequality and the boundedness of (u,v,w) asserted by Theorem 1.1. Thus we deduce from (4.74) that
E′3(t)≤−ε3C22E23(t)for allt>T3, | (4.76) |
which implies
E3(t)≤C3t−T3for allt>T3, | (4.77) |
with some positive constant C3. Hence we infer from the first inequalities in (4.71)–(4.73) that there exists a C4>0 such that
∫Ωu2dx+∫Ωv2dx+∫Ω(w−1)2dx≤C4E3(t)≤C3C4t−T3for allt>T3. | (4.78) |
It follows from the Gagliardo-Nirenberg inequality in two dimensions, Lemma 4.2 and Lemma 3.2 that there exist positive constants C5 and C6 such that
||u||L∞(Ω)+||v||L∞(Ω)+||w−1||L∞(Ω)≤C5(||u||23W1,4(Ω)||u||13L2(Ω)+||v||23W1,4(Ω)||v||13L2(Ω)+||w−1||23W1,4(Ω)||w−1||13L2(Ω))≤C6(∫Ωu2dx+∫Ωv2dx+∫Ω(w−1)2dx)16≤C6(C3C4)16(t−T3)−16 | (4.79) |
for all t>T3.
(b) When γi<θ, i=1,2, by means of the definitions of E3(t) and F∗3(t), it follows from the second inequalities in (4.71)–(4.73) that there exists a positive constant C7 such that
E3(t)≤C7(F∗3(t)+∫Ωudx+∫Ωvdx), | (4.80) |
for all t>T3.
By combining (4.70) with (4.80), we derive
ddtE3(t)≤−ε3C7E3(t)−1γ1(θ−γ1−ε3γ1)∫Ωudx−1γ2(θ−γ2−ε3γ2)∫Ωvdx | (4.81) |
for all t>0. Since γi<θ, i=1,2, we can select ε3≤min{1γ1(θ−γ1),1γ2(θ−γ2)} such that
ddtE3(t)≤−ε3C7E3(t)for all t > T_{3} , | (4.82) |
which means that there exist C8>0 and C9>0 satisfying
E3(t)≤C8e−C9(t−T3)for allt>T3. | (4.83) |
Thus we deduce from the first inequalities in (4.71)–(4.73) that there exists a C10>0 such that
∫Ωu2dx+∫Ωv2dx+∫Ω(w−1)2dx≤C10E2(t)≤C8C10e−C9(t−T3)for allt>T3. | (4.84) |
It follows from the Gagliardo-Nirenberg inequality in two dimensions, Lemma 4.2 and Lemma 3.2 that there exist positive constants C11 and C12 such that
||u||L∞(Ω)+||v||L∞(Ω)+||w−1||L∞(Ω)≤C11(||u||23W1,4(Ω)||u||13L2(Ω)+||v||23W1,4(Ω)||v||13L2(Ω)+||w−1||23W1,4(Ω)||w−1||13L2(Ω))≤C12(∫Ωu2dx+∫Ωv2dx+∫Ω(w−1)2dx)16≤C12(C8C10)16e−C9(t−T3)6 | (4.85) |
for all t>T3. The proof of Lemma 4.12 is complete.
Proof of Theorem 1.5. The statement of Theorem 1.5 is a direct consequence of Lemma 4.12.
The author is grateful to the editor and three anonymous referees for their valuable comments, which greatly improved the exposition of our paper. The author is also deeply grateful to Professor Renjun Duan for the support and friendly hospitality in CUHK. The work was partially supported by the National Natural Science Foundation of China (Grant Nos: 11601053, 11526042), Natural Science Foundation of Chongqing (Grant No. cstc2019jcyj-msxmX0082), Chongqing Municipal Education Commission Science and Technology Research Project (Grant No. KJZD-K202200602), China-South Africa Young Scientist Exchange Project in 2020, The Hong Kong Scholars Program (Grant Nos: XJ2021042, 2021-005) and Young Hundred Talents Program of CQUPT from 2022–2024.
The author declares that there is no conflict of interest.
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