Research article Special Issues

A mathematical model between keystone species: Bears, salmon and vegetation


  • We study an ecosystem of three keystone species: salmon, bears and vegetation. Bears consume salmon and vegetation for energy and nutrient intake, but the food quality differs significantly due to the nutritional level difference between salmon and vegetation. We propose a stoichiometric predator-prey model that not only tracks the energy flow from one trophic level to another but also nutrient recycling in the system. Analytical results show that bears may coexist with salmon and vegetation at a steady state, but the abundance of salmon may differ under different regimes. Numerical simulations reveal that a smaller vegetation growth rate may drive the vegetation population to extinction, whereas a large vegetation growth rate may drive the salmon population to extinction. Moreover, a large vegetation growth rate may stabilize the system where the bear, salmon and vegetation populations oscillate periodically.

    Citation: Christopher Middlebrook, Xiaoying Wang. A mathematical model between keystone species: Bears, salmon and vegetation[J]. Mathematical Biosciences and Engineering, 2023, 20(9): 16628-16647. doi: 10.3934/mbe.2023740

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  • We study an ecosystem of three keystone species: salmon, bears and vegetation. Bears consume salmon and vegetation for energy and nutrient intake, but the food quality differs significantly due to the nutritional level difference between salmon and vegetation. We propose a stoichiometric predator-prey model that not only tracks the energy flow from one trophic level to another but also nutrient recycling in the system. Analytical results show that bears may coexist with salmon and vegetation at a steady state, but the abundance of salmon may differ under different regimes. Numerical simulations reveal that a smaller vegetation growth rate may drive the vegetation population to extinction, whereas a large vegetation growth rate may drive the salmon population to extinction. Moreover, a large vegetation growth rate may stabilize the system where the bear, salmon and vegetation populations oscillate periodically.



    A keystone species is one on which other species in the ecosystem largely depend. So much so that if removed there are drastic effects on the ecosystem [1]. Bears, salmon and vegetation are unique in an ecosystem because all three are keystone species dependent on one another [2]. In this ecosystem, salmon and vegetation are both food resources for bears, but the food quality differs significantly.

    Salmon are much more nutrient-rich compared to vegetation, i.e., the nitrogen/phosphorous etc. concentration within salmon's bodies is much higher compared to vegetation. However, salmon must maintain homeostasis of element concentration within the body so that they must excrete excessive nutrients into the environment if the element concentration exceeds a threshold. On the other side, vegetation is less nutrient-rich but can absorb the remaining nutrient in the environment so the nutrient level within vegetation may differ significantly over time at an order of different magnitude [3,4,5].

    Bears share common characteristics with salmon where bears are also nutrient-rich but must maintain a homeostasis of element concentration within their bodies [3,4,5]. When consuming poor-quality food where the element concentration is low, the energy conversion from food consumption to the bear biomass cannot be maximized. In the bear-salmon-vegetation ecosystem, salmon is good-quality food, whereas vegetation is poor-quality food. Therefore, when studying the bear-salmon-vegetation ecosystem, it is important to devise a model that depicts not only the energy transfer but also the nutrient flow between trophic levels.

    One of the well-known models that study stoichiometric population dynamics is the one proposed by Loladze and Kuang [6]. The model builds on classical predator-prey models of one consumer and one food resource but incorporates a single limiting element phosphorus into modeling to distinguish food quantity and food quality. Following [6], stoichiometric models have been studied extensively, see [7,8,9,10,11,12,13] for example. More recently, Phan, Elser and Kuang extended the previous producer-grazer model framework by including multiple shared limiting elements in the modeling [14]. The formulated model excludes the non-smoothness in the original stoichiometric model but replicates qualitatively similar dynamics in a wide range of parameters [14]. We follow the modeling framework of the aforementioned studies but extend the model to include a specific nutrient-rich food resource, which agrees with the bear-salmon-vegetation ecosystem. In particular, we choose nitrogen as the limiting element of the system because nitrogen is vital for all salmon, bears and vegetation.

    We organize the paper in the following. In Section 2, we formulate a stoichiometric model of one consumer and two food resources, where one food resource is of good quality but the other food resource is of poor quality. In Section 3, we analyze the existence and local stability of the steady-state solutions. In Section 4, we conduct numerical simulations that confirm the analytical results. Moreover, numerical simulations also reveal interesting dynamics when certain parameters vary in a range. We end this paper in Section 5 with conclusions and discussions.

    In the bear-salmon-vegetation ecosystem, salmon and vegetation are both food resources for bears but bears do not consume certain organs of salmon. Rather, bears leave salmon corpses around vegetation so that the vegetation absorbs any remaining nitrogen that ultimately contributes to the growth of its population. Moreover, bears and salmon are much more nutrient-rich compared to vegetation. However, salmon and bears must maintain homeostasis in nitrogen concentration within their bodies and excrete excessive nitrogen into the environment. Vegetation is less nutrient-rich but can absorb the remaining nitrogen in the environment so the nitrogen concentration in vegetation may vary at an order of different magnitude.

    We first formulate a model that includes the energy transfer among trophic levels when nitrogen is abundant among all three species. Denote S(t),V(t),B(t) by the salmon population density (measured in carbon per litre), the vegetation population density (measured in carbon per litre) and the bear population density (measured in carbon per litre) respectively. The model is

    S(t)=αSf(S)B,V(t)=γV[1VK]g(V)B+ϵf(S)B+ψB,B(t)=e1f(S)B+e2g(V)BλB, (2.1)

    where α is the growth rate of the salmon population, f(S) is the consumption rate of salmon by the bear, γ is the intrinsic growth rate of the vegetation, K is the carrying capacity of the environment, g(V) represents the consumption rate of vegetation by the bear, e1 and e2 are the energy conversion rate from the consumption of the salmon and the vegetation respectively and λ is the natural death rate of the bear. In (2.1), ϵ represents the rate of contribution to the vegetation growth due to the element recycling from the nearby salmon corpses left by bears. Furthermore, the bears produce excrement after digesting the salmon, which provides another source for vegetation to absorb the nitrogen, which contributes positively to vegetation growth at a rate of ψ [2].

    The above model (2.1) assumes a linear growth rate for the salmon and a logistic growth for the vegetation. Having spent most of their lives feeding and growing at sea, salmon return to spawn and die. Salmons return to their natal streams carrying marine-derived nutrients in their body tissues. In particular, adult salmon are rich in nitrogen [2]. Salmon do not compete for resources which implies it is safe to assume a linear growth rate of γ. Vegetation, on the other hand, competes for resources limited by solar energy and follows a logistic growth rate. However, the vegetation population is not only limited by light but is also regulated by nitrogen concentration.

    Because of the homeostasis of bears and salmon, we can safely assume that the N: C (nitrogen: carbon) ratio within bears and salmon are θ1 and θ2 respectively. It follows that the remaining nitrogen in the environment is Nθ1B(t)θ2S(t). Moreover, following [6], we assume that the N: C (nitrogen: carbon) ratio within the vegetation never falls below a level NV. Combining the assumptions, we obtain another carrying capacity of the vegetation limited by nitrogen as

    Nθ1B(t)θ2S(t)NV.

    Together with the carrying capacity limited by solar energy, we obtain an improved carrying capacity for the vegetation population

    min{K,Nθ1B(t)θ2S(t)NV}.

    Following [6], we assume that all of the nitrogen within the system gets recycled immediately and there is no pool of free nitrogen in the environment.

    It follows that the N: C ratio in vegetation is

    Nθ1B(t)θ2S(t)V(t).

    Vegetation serves as good-quality food for bears if

    Nθ1B(t)θ2S(t)V(t)θ1,

    or equivalently

    Nθ1B(t)θ2S(t)θ1V(t)1,

    and is poor-quality food otherwise. If vegetation is good-quality food, the biomass conversion from the vegetation to the bear follows the maximum energy intake e2. However, if vegetation is poor-quality food, the biomass conversion is reduced by a ratio

    Nθ1B(t)θ2S(t)θ1V(t).

    Hence, biomass conversion efficiency is not a constant but depends on both energetic and nutrient limitations. Note that we can assume that salmon is a good-quality food source for bears since salmon is nutrient-rich. Following [6], we formulate the biomass conversion from vegetation consumption as

    e2min{1,Nθ1B(t)θ2S(t)θ1V(t)}.

    Taking all the aforementioned evidence into consideration, we arrive at the model below

    S(t)=αSf(S)B,V(t)=γV[1Vmin{K,Nθ1Bθ2SNV}]g(V)B+ϵf(S)B+ψB,B(t)=e1f(S)B+e2min{1,Nθ1Bθ2Sθ1V}g(V)BλB, (2.2)

    where the parameters are listed in Table 1.

    Table 1.  Parameters of model (2.2).
    Parameter Description Restrictions
    α Growth rate of salmon α>0
    β Predation rate on salmon β>0
    γ Intrinsic growth rate for vegetation γ>0
    K Carrying capacity of vegetation limited by solar energy K>0
    N Total mass of nitrogen within the system N>0
    θ1 N: C ratio in bears θ1>0
    θ2 N: C ratio in salmon θ2>0
    NV Minimum N:C ratio in vegetation NV>0
    δ Predation rate on vegetation δ>0
    ϵ Contribution to vegetation growth from dead salmon organ ϵ0
    ψ Contribution to vegetation growth from bear excretion ψ0
    e1 Maximum biomass conversion efficiency via salmon consumption 1>e1>0
    e2 Maximum biomass conversion efficiency via vegetation consumption 1>e2>0
    λ Natural mortality rate of bears λ>0

     | Show Table
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    In the following analysis, we assume a linear functional response for both predation of the salmon and the vegetation, i.e., f(S)=βS and g(V)=δV. We analyze a special case where the element recycling for the vegetation is minimum and can be ignored, i.e., ϵ=0 and ψ=0.

    In the following analysis, we restrict the positive invariant set Ω to be

    Ω={(S,V,B):S>0,V>0,B>0,θ1B+θ2S+NVV<N},

    where the population densities are bounded by the total element concentration N in the ecosystem.

    We first analyze steady-state solutions of (2.2) when ϵ=0,ψ=0. The steady-state solutions are determined by

    αSβSB=0, (3.1)
    γV[1Vmin{K,Nθ1Bθ2SNV}]δVB=0, (3.2)
    e1βSB+e2min{1,Nθ1Bθ2Sθ1V}δVBλB=0. (3.3)

    Here, (3.1) implies that S=0 or B=α/β. If salmon is at extinction, i.e., S=0, then (3.3) implies that

    B=0ore2min{1,Nθ1Bθ1V}δVλ=0.

    If the bear population is at extinction, i.e., B=0, then (3.2) simplifies to

    γV[1Vmin{K,N/NV}]=0.

    Therefore, we obtain an extinction equilibrium E0(0,0,0) and a vegetation-only equilibrium E1(0,min{K,N/NV},0) that always exist.

    If

    e2min{1,Nθ1Bθ1V}δVλ=0,

    then we have

    V=λe2δ,ifδVδθ1(Nθ1B), (3.4)

    or

    ˜B=Nθ1λe2δ,ifδV>δθ1(Nθ1B). (3.5)

    When (3.4) holds, (3.2) reduces to

    γV[1Vmin{K,Nθ1BNV}]δVB=0. (3.6)

    If K<(Nθ1B)/NV, then (3.6) leads to

    B=γδ(1VK). (3.7)

    Therefore, a boundary equilibrium E2(0,V,B) exists if

    {B<min{Nθ1λe2δ,Nθ1KNVθ1},V<K. (3.8)

    If K>(Nθ1B)/NV, then (3.6) simplifies to

    γVγNV(V)2Nθ1B=δVB, (3.9)

    which is equivalent to

    Ω1B2+Ω2B+Ω3=0,

    where Ω1=θ1δV,Ω2=V(θ1γ+δN),Ω3=γV(NNVV). By the invariant set Ω, we have B<N/θ1. We solve for B and reject the larger root to obtain

    B=Ω2Ω224Ω1Ω32Ω1.

    Therefore, a boundary equilibrium E3(0,V,B) exists if

    Nθ1KNVθ1<B<Nθ1λe2δ.

    Finally, when (3.5) holds, (3.2) reduces to

    γV[1Vmin{K,Nθ1˜BNV}]δV˜B=0. (3.10)

    Solving for V and rejecting the trivial solution leads to

    ˜V=γδ˜Bγmin{K,Nθ1˜BNV}.

    Therefore, a boundary equilibrium E4(0,˜V,˜B) exists if

    Nθ1>λe2δ,˜V>λe2δ.

    Next, we analyze the existence of steady-state solutions when the bear population exists, i.e., ˆB=α/β. Substituting ˆB into (3.3) gives

    e1βˆBS+e2min{δˆBV,δˆBθ1(Nθ1ˆBθ2S)}λˆB=0. (3.11)

    If δˆBV>[δˆB(Nθ1ˆBθ2S)]/θ1, solving for S in (3.11) leads to

    ˉS=θ1λe2δ(Nθ1ˆB)e1βθ1e2δθ2. (3.12)

    We further substitute ˉS into (3.2) and obtain

    γV[1Vmin{K,(Nθ1ˆBθ2ˉS)/NV}]δˆBV=0. (3.13)

    By solving for V and denoting the positive solution by ˉV, we obtain

    ˉV=min{K,Nθ1ˆBθ2ˉSNV}(1δˆBγ).

    Therefore, a positive equilibrium E5(ˉS,ˉV,ˆB) exists if

    θ1λe2δ(Nθ1ˆB)e1βθ1e2δθ2>0,ˉV>Nθ1ˆBθ2ˉSθ1.

    If δˆBV<[δˆB(Nθ1ˆBθ2S)]/θ1, (3.3) simplifies to

    e1βS+e2δVλ=0.

    Solving for S gives

    S=λe2δVe1β. (3.14)

    We substitute (3.14) into (3.2) and obtain

    γV[1Vmin{K,[Nθ1ˆBθ2(λe2δV)/(e1β)]/NV}]δˆBV=0. (3.15)

    If K<(Nθ1ˆB)/NV[θ2(λe2δV)]/(e1βNV), or equivalently [e1βKNV(Nθ1ˆB)e1β+λθ2]/[e2δθ2]<V, then (3.15) reduces to

    γV(1VK)δˆBV=0.

    Solving for V and denoting the positive solution by ˆV gives

    ˆV=Kγ(γδˆB). (3.16)

    Therefore, a positive equilibrium E6(ˆS,ˆV,ˆB) exists if

    {(θ1e1βθ2e2δ)ˆV<e1βNθ2λθ1e1βˆB,ˆV<λe2δ,e1βKNV(Nθ1ˆB)e1β+λθ2e2δθ2<ˆV.

    If K>(Nθ1ˆB)/NV[θ2(λe2δV)]/(e1βNV), or equivalently V<[e1βKNV(Nθ1ˆB)e1β+λθ2]/[e2δθ2], then (3.15) reduces to

    γV[1V(Nθ1ˆB)/NVθ2(λe2δV)/(e1βNV)]ˆBδV=0. (3.17)

    Solving for V and denoting the positive solution by V+ leads to

    V+=(γˆBδ)[e1β(Nθ1ˆB)θ2λ]γe1βNV(γˆBδ)θ2e2δ.

    Therefore, a positive equilibrium E7(S+,V+,ˆB) exists if

    {θ1ˆB+θ2S++θ1V+<N,V+<min{λe2δ,e1βKNV(Nθ1ˆB)e1β+λθ2e2δθ2}.

    Next, we analyze the local stability of the steady-state solutions. Direct calculations lead to the Jacobian matrix

    J=(αβB0βSJ21J22J23J31J32J33), (3.18)

    where

    J21={0,ifK<Nθ1Bθ2SNV,γθ2NVV2(Nθ1Bθ2S)2,ifK>Nθ1Bθ2SNV,J22={γ2γVKδB,ifK<Nθ1Bθ2SNV,γ2γNVVNθ1Bθ2SδB,ifK>Nθ1Bθ2SNV,J23={δV,ifK<Nθ1Bθ2SNV,γθ1NVV2(Nθ1Bθ2S)2δV,ifK>Nθ1Bθ2SNV,J31={e1βB,if1<Nθ1Bθ2Sθ1V,e1βBe2δθ2θ1B,if1>Nθ1Bθ2Sθ1V,J32={e2δB,if1<Nθ1Bθ2Sθ1V,0,if1>Nθ1Bθ2Sθ1V,
    J33={e1βS+e2δVλ,if1<Nθ1Bθ2Sθ1V,e1βS+e2δθ1(N2θ1Bθ2S)λ,1>Nθ1Bθ2Sθ1V.

    Then, the characteristic equation follows as |μIJ|=0. By substituting each equilibrium into the characteristic equation, we are able to obtain the following theorems that state the stability results. The theorem below shows the stability of the bear-extinction equilibria.

    Theorem 3.1. The trivial equilibrium E0(0,0,0) and the vegetation-only equilibrium E1(0,min{K,N/NV},0) are unstable.

    Proof. By evaluating (3.18) at the trivial equilibrium E0 and solving the characteristic equation |μIJ|=0, we obtain the characteristic roots

    μ1=α,μ2=γ,μ3=λ,

    which demonstrates that E0 is unstable.

    Similar calculations give the characteristic roots at E1

    μ1=α,μ2=γ,μ3={e2δmin{K,NNV}λ,ifθ1min{K,NNV}<N,e2δθ1Nλ,ifθ1min{K,NNV}>N,

    which shows that E1 is unstable.

    The following theorem shows the stability results of the salmon-extinction equilibria E2,E3,E4.

    Theorem 3.2. The salmon-extinction equilibrium E2 is locally asymptotically stable if α/β<B and is unstable if otherwise. The salmon-extinction equilibrium E3 is locally asymptotically stable if α/β<B<γ/δ and is unstable if otherwise. The salmon-extinction equilibrium E4 is locally asymptotically stable if α/β<˜B<γ/δ and is unstable if otherwise.

    Proof. Direct calculations lead to the characteristic equation at E2

    (μ(αβB))(μ2J22μJ23J32)=0,

    where J22=γ(2γV)/KδB,J23=δV,J32=e2δB. By using (3.7), we can simplify J22=γ(2γV)/KδB=γ(2γV)/Kγ(1V/K)=(γV)/K<0. It follows that μ1=αβB,μ2+μ3=(γV)/K<0,μ2μ3=δVe2δB>0. Therefore, E2 is locally asymptotically stable if μ1<0, i.e., (α/β)<B and is unstable if otherwise.

    Similarly, by evaluating (3.18) at E3, we obtain the characteristic equation at E3

    (μ(αβB))(μ2J22μJ23J32)=0,

    where

    J22=γ2γNVVNθ1BδB,J23=γθ1NV(V)2(Nθ1B)2δV,J32=e2δB.

    This implies that μ1=αβB,μ2+μ3=J22,μ2μ3=J23J32>0. By using (3.9), we can further simplify J22=γ2(γδB)δB=δBγ. Therefore, E3 is locally asymptotically stable if μ1<0 and μ2+μ3<0, which are equivalent to α/β<B<γ/δ.

    Finally, direct calculations give the characteristic equation at E4

    (μ(αβ˜B))(μJ22)(μJ33)=0,

    where

    J22={γ2γ˜VKδ˜B,ifK<Nθ1BNV,γ2γNV˜VNθ1˜Bδ˜B,ifK>Nθ1BNV,J33=λe2δNθ1.

    We can further simplify J22 by using (3.10) to obtain J22=δ˜Bγ. Moreover, J33<0 is satisfied automatically when E4 exists. Therefore, E4 is locally asymptotically stable if α/β<˜B<γ/δ and is unstable otherwise.

    Finally, the following theorem shows the local stabilities of the positive equilibria E5,E6,E7.

    Theorem 3.3. The positive equilibrium E5(ˉS,ˉV,ˆB) is locally asymptotically stable if e1βθ1>e2δθ2 and is unstable if otherwise. The positive equilibrium E6(ˆS,ˆV,ˆB) is locally asymptotically stable if

    e2δ(δβ)K(γδˆB)γ+βλ>0

    and is unstable if otherwise. The positive equilibrium E7(S+,V+,ˆB) is always unstable.

    Proof. Direct calculations give the characteristic equation at E5

    (μJ22)(μ2J33μJ13J31)=0,

    where

    J22={γ2γˉVKδˆB,ifK<Nθ1ˆBθ2ˉSNV,γ2γNVˉVNθ1ˆBθ2ˉSδˆB,ifK>Nθ1ˆBθ2ˉSNV,J33=e1βˉS+e2δθ1(N2θ1ˆBθ2ˉS)λ,J13=βˉS,J31=ˆBe1βθ1e2δθ2θ1.

    By using (3.12), we can simplify J33 as J33=e2δˆB<0. Moreover, by using (3.13), we can simplify J22 to

    J22={γˉVK<0,ifK<Nθ1ˆBθ2ˉSNV,γNVˉVNθ1ˆBθ2ˉS<0,ifK>Nθ1ˆBθ2ˉSNV.

    The above analyses show that E5 is locally asymptotically stable if J31>0 or equivalently e1βθ1>e2δθ2.

    Similarly, by evaluating (3.18) at E6, we obtain the characteristic equation

    μ3(J22+J33)μ2+(J22J33J23J32J13J31)μ+J13J31J22=0,

    where

    J13=βˆS<0,J22=γ2γˆVKδˆB,J23=δˆV<0,J31=e1βˆB>0,J32=e2δˆB>0,J33=e1βˆS+e2δˆVλ.

    By the Routh-Hurwitz stability criterion, E6 is locally asymptotically stable if

    {(J22+J33)>0,J22J33J23J32J13J31>0,J13J31J22>0,(J22+J33)(J22J33J23J32J13J31)>J13J31J22. (3.19)

    Here we can simplify J22 and J33 by using (3.16) and (3.14) respectively to obtain

    J22=γˆVK<0,J33=0,

    which further reduce (3.19) to

    {J22>0,J23J32+J13J31<0,J13J31J22>0,J22J23J32>0.

    Therefore, E6 is locally asymptotically stable if J23J32+J13J31<0 or equivalently

    e2δ(δβ)K(γδˆB)γ+βλ>0.

    Finally, similar to the above analyses, we obtain the characteristic equation at E7

    μ3J22μ2(J23J32+J13J31)μ+J13(J31J22J21J32)=0,

    where

    J13=βS+<0,J21=γθ2NVV2(Nθ1Bθ2S)2<0,J22=γ2γNVVNθ1Bθ2SδB,J23=γθ1NVV2(Nθ1Bθ2S)2δV<0,J31=e1βˆB>0,J32=e2δˆB>0,J33=e1βS++e2δV+λ.

    By the Routh-Hurwitz criterion, E7 is locally asymptotically stable if

    {J22>0,(J23J32+J13J31)>0,J13(J31J22J21J32)>0,J22(J23J32+J13J31)>J13(J31J22J21J32).

    Here it is obvious that (J23J32+J13J31)>0. Next, by substituting (3.14) and (3.17) into J22, we can simplify

    J22=γ2γNVVNθ1Bθ2SδB=e1βNVγVe1βNe1βθ1ˆBθ2λ+θ2e2δV2γNVVNθ1ˆBθ2S<0,

    which is satisfied when E7 exists. Moreover, J22(J23J32+J13J31)>J13(J31J22J21J32) is equivalent to J22J23+J13J21>0. It remains to verify that J13(J31J22J21J32)>0 or equivalently J31J22J21J32<0. Here J31J22J21J32<0 can be simplified to 0<γθ2NVe2δV2 by substituting (3.14) and (3.17), which leads to the contradiction. Therefore, E7 is always unstable when exists. Thus, completes the proof.

    Now we explore the dynamics of (2.2) numerically. For the numerical simulation, we will be assuming that the salmon-bear-vegetation ecological system lies within the riparian forests of Alaska. More specifically, we will be looking at a 100km2 region of Lynx Creek, a tributary of the Wood River Lakes system in the Bristol Bay region of southwestern Alaska, USA (5929N,15855W) [2]. This is a well-documented region where we will be able to get accurate values for our parameters. Based on data from [2], we explore the following scenarios by using biologically realistic parameters.

    Figure 1 shows that the salmon-extinction boundary equilibrium is locally asymptotically stable. In particular, 1(a) demonstrates that E2 is locally asymptotically stable, whereas both E3,E4 do not exist under the parameter set. Figure 1(b) indicates that E3 is locally asymptotically stable, whereas both E2,E4 do not exist under the parameter set. Moreover, Figure 1(c) shows that E4 is locally asymptotically stable, whereas both E2,E3 do not exist under the parameter set. Via our extensive numerical experiments, we find that E2 cannot coexist with either/both E3,E4. Figure 2 demonstrates that all salmon, vegetation and bear populations coexist together. Figure 2(a) shows that the positive equilibrium E5 is locally asymptotically stable, whereas E6 does not exist under the parameter set. Figure 2(b) shows that the positive equilibrium E6 is locally asymptotically stable, whereas E5 does not exist under the parameter set.

    Figure 1.  Locally asymptotically stable E2,E3,E4 respectively. Parameter values for 1(a): α=0.008,β=0.3507,γ=1.1036,K=6.8951,N=0.9533,θ1=0.2371, θ2=0.1371,NV=0.0115,δ=0.6217, e1=0.5676,e2=0.8817,λ=0.5088. Parameter values for 1(b): α=0.9746, β=0.6764,γ=1.4623,K=35.4971,N=0.9610,θ1=0.2046,θ2=0.1046,NV=0.0367, δ=0.7535,e1=0.4830,e2=0.3214,λ=0.6319. Parameter values for 1(c): α=0.8815,β=0.9483,γ=9.7637,K=33.1267,N=0.6990,θ1=0.2623,θ2=0.1623, NV=0.0873,δ=0.8055,e1=0.1425,e2=0.1705,λ=0.1755.
    Figure 2.  Locally asymptotically stable E5,E6 respectively. Parameter values for 2(a): α=0.3860,β=0.8874,γ=4.7299,K=18.1625,N=0.9595,θ1=0.5330, θ2=0.4330,NV=0.0720,δ=0.6159, e1=0.8917,e2=0.2849,λ=0.7341. Parameter values for 2(b): α=0.5441,β=0.9575,γ=0.9652,K=1.6234,N=0.9988,θ1=0.6647, θ2=0.5647,NV=0.0758,δ=0.9076,e1=0.8909,e2=0.4548,λ=0.3478.

    Next, we explore how the vegetation growth rate impacts the long-term dynamics of (2.2). We generate Figure 3 by using the same set of parameter as Figure 2(b) but varying γ. Figure 3(a) demonstrates that a small vegetation growth rate drives the vegetation population to extinction, whereas salmon coexist with bears, but the populations oscillate periodically. On the other hand, the salmon population goes to extinction, but the vegetation coexists with bears at a steady state when γ is large. The results are reasonable biologically because a small vegetation growth rate cannot sustain the persistence of the vegetation population. However, a large vegetation growth rate facilitates the nitrogen recycling of the vegetation, which supports the bear population via consumption but drives the salmon population to extinction due to the scarcity of nitrogen.

    Figure 3.  Salmon, vegetation and bear populations over time with small γ (3(a)) or large γ (3(b)). The parameters are the same as Figure 2(b) except γ, where γ=0.5 for 3(a) and γ=1.2 for 3(b).

    Finally, Figure 4(a) demonstrates another scenario where the salmon, bear and vegetation coexist but the populations oscillate periodically. However, Figure 4(b) shows that a larger vegetation growth rate may stabilize the oscillating populations to a steady state.

    Figure 4.  Salmon, vegetation and bear populations over time with small γ (4(a)) or large γ (4(b)). The parameters values are α=2,β=1.2,K=15,N=1.2,θ1=0.1,θ2=0.15,NV=0.1, δ=0.9,e1=0.8,e2=0.1,λ=1.8,γ=1.6 for 4(a), and γ=4 for 4(b).

    Next, we explore the extended model of (2.2) where the salmon population follows a logistic growth

    S(t)=αS(1SK1)f(S)B.

    We consider that the predation of the salmon and the vegetation now follow the Holling type Ⅱ functional response [15]

    f(S)=βSh1+S,g(V)=δVh2+V,

    where h1,h2 represent the half-saturating constants respectively.

    Figure 5(a) demonstrates the bifurcation diagram of the salmon population with respect to the salmon carrying capacity K1. Moreover, Figure 5(b) demonstrates the bifurcation diagram of the vegetation population with respect to K1. The bifurcation diagram of the bear population with respect to K1 is similar to Figure 5(a) and is thus omitted. Figure 5 shows that the salmon, vegetation and bear populations coexist at a steady state if the salmon carrying capacity is relatively small. However, when K1 increases and passes 4.1, the vegetation population goes to extinction, whereas the salmon population coexists with the bear population at a steady state. Moreover, when K1 further increases and passes 5.7, the vegetation population remains at extinction, but the salmon and bear populations oscillate periodically. The results are biologically reasonable because if the salmon carrying capacity is relatively small, the salmon, vegetation and bear compete for the nutrients but coexist due to the scarcity of the salmon population. However, if the salmon carrying capacity is at the intermediate range, the competition for nutrients drives the vegetation population to extinction because the salmon and bear are more nutrient-rich. Finally, a large salmon carrying capacity destabilizes the coexistence steady-state of the salmon and bear due to the enrichment of salmon.

    Figure 5.  Bifurcation diagram of the salmon population/vegetation population with respect to the salmon carrying capacity K1 for 5(a) and 5(b) respectively. The parameters values are the same as 3(a) except β=4.7875,δ=4.538,h1=5,h2=5.

    Next, to compare the dynamics of the salmon-vegetation-bear model with the grazer-consumer model in [6], we run the simulation by using the same set of parameters as in [6], except the salmon carrying capacity K1 and the half-saturating constant of the salmon consumption h1. Figure 6 shows the bifurcation diagram of the salmon population/vegetation population/bear population with respect to the vegetation carrying capacity K respectively. If K is relatively small, Figure 6 shows that the salmon, bear and vegetation populations coexist at a steady state. However, when K increases and passes 0.24, the salmon population goes to extinction, whereas the vegetation and bear populations coexist at a steady state. If K further increases and passes 0.55, the vegetation and bear coexist but the populations oscillate periodically. If K further increases and passes 0.97, the vegetation and bear populations coexist but return to the steady state status. Finally, if K becomes relatively large and passes 1.5, the salmon population is no longer extinct but coexists with the vegetation and bear populations at a steady state.

    Figure 6.  Bifurcation diagram of the salmon population/vegetation population/bear population with respect to the vegetation carrying capacity K for 6(a), 6(b) and 6(c) respectively. Parameters values: α=0.8,β=0.95,γ=1.2,N=0.025,θ1=0.03,θ2=0.03,NV=0.0038, δ=0.81,e1=0.9,e2=0.8,λ=0.25,K1=0.3,h1=0.25,h2=0.25.

    Overall, when K is within the intermediate range, i.e., the salmon population is at extinction, Figures 6(a), 6(b) and 6(c) demonstrate similar dynamics with the producer-grazer model in [6]. The vegetation-bear coexistent steady state loses stability due to energy enrichment. However, a larger carrying capacity limited by the light energy drives the oscillating vegetation and bear populations to a steady state because of the competition for nutrients. Different from the results in [6], the bear population does not go to extinction but coexists with either/both salmon or/and vegetation populations when K is relatively small or large. The results suggest that a low or high carrying capacity limited by light energy facilitates the persistence of all salmon, vegetation and bear populations. Moreover, an intermediate carrying capacity limited by light energy drives the salmon population to extinction.

    In this paper, we study an ecosystem of bears, salmon and vegetation, where bears consume salmon and vegetation for survival. Because salmon return to their natal streams to spawn and carry marine-derived nutrients, it is important to characterize the nutritional level difference between salmon, bears, and vegetation. In general, salmon and bears are more nutrient-rich compared to vegetation but must maintain homeostasis of the nutritional level within their bodies. On the other side, vegetation is less nutrient-rich but can recycle the remaining nutrient in the environment and the nutritional level within the body may differ significantly.

    We propose a stoichiometric predator-prey model that tracks both the energy flow and the nutrient recycling from one trophic level to another. Analytical results show that boundary equilibria E0,E1 where bears are extinct exist but are always unstable. Moreover, boundary equilibria E2,E3,E4 where salmon are extinct but bears persist may exist and remain locally asymptotically stable if certain conditions are satisfied. Positive equilibria E5,E6,E7 where salmon, bears and vegetation coexist may exist if certain conditions are satisfied. Analyses show that E5 and E6 may remain locally asymptotically stable under certain conditions, but E7 is always unstable.

    Numerical simulations demonstrate that a small vegetation growth rate may drive the vegetation population to extinction where the salmon population and the bear population coexist in the periodic setting. Moreover, a large vegetation growth rate may drive the salmon population to extinction but the vegetation coexists with the bears at a steady state. Alternatively, the salmon, bears and vegetation populations may coexist periodically. In this scenario, a large vegetation growth rate may stabilize the system and drive the salmon, bear and vegetation to coexist at a steady state.

    In this paper, for the analytical analysis, we adopt the linear functional response for the predation of the salmon and the predation of the vegetation in (2.2) for simplicity. However, a linear functional response has its limitation and is more suitable for an ecosystem with a sparse population density. A Holling type Ⅱ functional response has a saturating effect when population density becomes large and therefore may be suitable for a broader regime [15]. Figures 5 and 6 in the simulation also confirm that rich dynamics may occur if the Holling type Ⅱ functional response is adopted. Because of the nonlinearity of the Holling type Ⅱ functional response, we expect that the analytical analysis becomes more challenging but on the other hand may deepen our understanding of the ecosystem of keystone species, which leaves as future works.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    CM is grateful for funding from the NSERC USRA. XW is grateful for funding from the NSERC of Canada (RGPIN-2020-06825 and DGECR-2020-00369).

    The authors declare there is no conflict of interest.



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