# Qualitative Analysis for a Reaction-Diffusion Predator-Prey Model with Disease in the Prey Species

Authors: Meihong Qiao; Anping Liu; Urszula Foryś
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

A diffusive predator-prey system with disease in predator species and no-flux boundary condition is considered. Sufficient conditions which ensure persistence of the system are obtained. Conditions of disease-free ecosystem are also studied. Furthermore, sufficient conditions for global asymptotic stability of the unique positive equilibrium and disease-free equilibrium of the system are derived using the approach of Lyapunov function.


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## Body

## 1. Introduction

Ecoepidemiology is a relatively new branch of study in theoretical biology, which tackles problems by dealing with both ecological and epidemiological approach. It can be viewed as the coupling of an ecological predator-prey or competition model with an epidemiological SI, SIS, or more complex model. Anderson and May [1] were the first who marked that the effect of disease in ecological systems is an important issue from both mathematical and ecological point of view. They proposed an ecoepidemiological model by merging the ecological predator-prey model introduced by Lotka and Volterra with epidemiological model.Clearly, in the natural world, species does not exist alone. While the disease is spread within the species, the species also competes with other species for environmental resources like space or food, or is predated by other species. Therefore, it is of more biological significance to consider the effect of interacting species when we study the dynamical behaviors of epidemiological models.Many papers have been devoted to study the effects of a disease on a predator-prey system. Venturino [2] studied SI and SIS models with disease spread among the prey when the logistic growth of both the prey and predator populations is assumed and the predators eat infected preys only. In [3], Hsu and Huang considered the following predator-prey model:
(1)dudt=ru(1-uK)-p(u)v,dvdt=sv(1-hvu),
where u and v represent densities of the populations of prey and predators, respectively, and r, s, K, and h are positive constants. The population of prey grows logistically with carrying capacity K and intrinsic growth rate r in the absence of predation. Predators consume prey according to the functional response p(u) and grow logistically with intrinsic growth rate s. Carrying capacity of the predator species is proportional to the size of the prey population. It should be noticed that the model described by (1) is a generalisation of the prey-predator model proposed by May [4] which is known as Holling-Tanner model. In this model the functional response p(u)=au/(b+u) is of the Holling type [5], and it is one of the prototype models involving limit cycle dynamics.


## 2. Model Formulation

On the basis of (1) we propose an ecoepidemiological model with a disease spread in the predator population. We assume that only predator can be infected and the infected individual does not recover or become immune. Because the predation ability of healthy (and susceptible at the same time) predators is stronger than infected ones, we suppose that prey can be preyed on only by healthy predators. Moreover, we also assume the simplest linear form of functional response p(u)=ku. Therefore, the model reads
(2)dudt=ru(1-uK)-kuv,dvdt=sv(1-hvu)-βwv,dwdt=βwv-dw-μw2,
where u, v, and w represent densities of the populations of prey, susceptible predator, and infected predator, respectively. The death rate of infected predators equals d, β is the infectious rate of the disease, and μ is the density-dependent death rate of infected predators. Other parameters are the same as in (1).Species dispersal is one of the most prevalent phenomena of nature, and many empirical studies and monographs on population dynamics in a spatial heterogeneous environment have been done (see [6–15] and the references cited therein). Most important subjects of population diffusion models are coexistence of populations, local and global stability of equilibria, existence of periodic solutions, and so forth, (see [16–20]). In particular, single population models were considered, for example, in [21–23], while predator-prey system with the prey dispersal was studied, for example, in [24–26]. Such type of model is still of great interest and importance; compare the recent papers [27–30] and the references therein.Taking into account inhomogeneous distribution of predators and their prey in different spatial locations within a fixed bounded domainΩ in ℝN with smooth boundary at any given time and the natural tendency of each species to diffuse to areas of smaller population density, we are led to consider the following reaction-diffusion system:
(3)∂u∂t=d1Δu+ru(1-uK)-kuv,x∈Ω,∂v∂t=d2Δv+sv(1-hvu)-βwv,x∈Ω,∂w∂t=d3Δw+βwv-dw-μw2,x∈Ω,∂u∂η=∂v∂η=∂w∂η=0,x∈∂Ω,u(x,0)=u0(x)≥0,v(x,0)=v0(x)≥0,w(x,0)=w0(x)≥0,x∈Ω,
where Ω is a bounded domain in ℝN (N=2 or 3 in reality) with smooth boundary ∂Ω, ∂/∂η is the outward derivative normal to ∂Ω, and d1, d2, and d3 are strictly positive diffusion coefficients.To reduce the number of parameters we make the following change of variables:(4)rt⟼t,uK⟼u,bv⟼v,μw⟼w,
and we rename parameters accordingly,
(5)kr⟼k,sr⟼s,shrK⟼b,d1r⟼d1,d2r⟼d2,d3r⟼d3,βr⟼β,dr⟼d,μr⟼μ,
obtaining nondimensional version of the model
(6)∂u∂t=d1Δu+u(1-u)-kuv,x∈Ω,∂v∂t=d2Δv+v(s-vu)-βwv,x∈Ω,∂w∂t=d3Δw+βwv-dw-w2,x∈Ω,∂u∂η=∂v∂η=∂w∂η=0,x∈∂Ω,u(x,0)=u0(x)≥0,v(x,0)=v0(x)≥0,w(x,0)=w0(x)≥0,x∈Ω.In this paper we assume thatN=1 as calculations are simpler in such a case. However, the results presented below can be extended for N>1. Our goal is to give conditions guaranteeing persistence of the ecosystem described by (6). The persistence means that the disease is spread and endemic equilibrium appears. Equations (6) have positive (endemic) equilibrium (u*,v*,w*), where
(7)u*=12β2(+(β2-kdβ-ks-1)2+4β2)β2-kdβ-ks-1+(β2-kdβ-ks-1)2+4β2),v*=1-u*k,w*=βv*-d.Notice thatv*>0⇔u*<1 and w*>0⇔v*>d/β⇔u*<1-dk/β, which means that β>dk is the necessary condition for the existence of the positive equilibrium.One can easily check thatu*<1, while the inequality v*>d/β is equivalent to β>dk+d/s.On the other hand, it is also important to know conditions for disease-free ecosystem. Assuming thatw¯=0 we obtain semitrivial disease-free equilibrium (u¯,v¯,0) with
(8)u¯=11+ks,v¯=su¯.Remark 1.
For any parameter values (6) have the semitrivial equilibrium DFE with coordinates (u¯,v¯,0).
Ifβ>dk+(d/s), then (6) have the unique positive equilibrium EE with coordinates (u*,v*,w*).In the next sections we focus on the analysis of (6) and propose conditions for global stability of EE. We start from the case without diffusion (di=0 for i=1, 2, 3) and then turn to the system with positive diffusion coefficients.


## 3. The Model without Diffusion

In this section we assumedi=0, i=1, 2, 3; that is, consider spatially homogenous case
(9)u˙=u(1-u)-kuv,v˙=v(s-vu)-βwv,w˙=βwv-dw-w2.It is obvious that local solutions of (9) exist and are unique for any positive initial data (u0,v0,w0). Moreover, if w0=0, then w(t)=0 for all t≥0. Hence, we assume that some infected predators appear at t=0 and we want to know if the disease spreads in the ecosystem.Basing on the positivity of solutions we obtain the following estimates:(10)u˙≤u(1-u)⟹u(t)≤uM:=max⁡{u0,1},
which yields
(11)v˙≤v(s-vuM)⟹v(t)≤vM:=max⁡{suM,v0},
and therefore
(12)w˙≤w(βvM-d-w)⟹w(t)≤wM:=max⁡{βvM-d,w0}.
Notice that if solutions are bounded, then their derivatives are bounded as well, and therefore solutions exist for all t≥0. Moreover, if u0≤1, v0≤s, and w0≤βs-d for d<βs, then u(t)≤1, v(t)≤s, and w(t)≤βs-d for all t≥0. Hence, 𝒟={(u,v,w):u≤1, v≤s, w≤βs-d}, βs>d, is positively invariant for (9).Moreover, we obtain(13)u˙≥u(1-u-ks),
and assuming ks<1 we obtain u(t)>δu for any u0≥δu, δu<1-ks, as solutions of the logistic equation are increasing below the carrying capacity threshold. Similarly,
(14)v˙≥v(s-vδu-β2s),
and if β<1, then v(t)>δv for v0≥δv, δv<s(1-β2)δu. Finally,
(15)w˙≥w(βδv-d-w),
implying that if d<βδv, then w(t)>δw for w0≥δw with δw<βδv.Corollary 2.
Ifβs>d, then the set 𝒟=[0,1]×[0,s]×[0,βs-d] is positively invariant and globally attractive, while if βs≤d, then 𝒟=[0,1]×[0,s]×[0,βs] is positively invariant and globally attractive for (9).
If additionallyks<1, then the set ℰ=[δ,1]×[0,s]×[0,βs-d], βs>d (or ℰ=[δ,1]×[0,s]×[0,βs], βs≤d), is positively invariant and globally attractive for any δ<1-ks.
Ifks<1, β<1, and d<βs(1-β2)(1-ks), then the set 𝒢=[δu,1]×[δv,s]×[δw,βs-d] is positively invariant and globally attractive for any δu<1-ks, δv<s(1-β2)δu, and δw<βδv-d.Corollary 3.
Ifks<1, β<1, and d<βs(1-β2)(1-ks), then the system described by (9) is persistent; that is, all solutions are bounded below from 0 and bounded above by some positive constants.


### 3.1. Local Stability of DFE and EE

Jacobi matrix for (9) reads
(16)J(u,v,w)=(1-2u2-kv-ku0v2u2s-2vu-βw-βv0βwβv-d-2w).
For DFE we have the following relations: 1-u¯-kv¯=0, v¯=su¯, and hence
(17)JDFE=(-u¯-ku¯0s2-s-βv¯00βv¯-d),
implying that local stability of DFE depends on the sign of βv¯-d, as submatrix (-u¯-ku¯s2-s) generates eigenvalues with negative real parts. We easily see that βv¯-d>0 is equivalent to β>dk+(d/s), that is, to the existence of EE, according to Remark 1.For EE we have the relations1-u*-kv*=0, s-(v*/u*)-βw*=0, and βv*-d-w*=0, and therefore
(18)JEE=(-u*-ku*0(v*u*)2-v*u*-βv*0βw*-w*).
Calculating characteristic polynomial for EE we obtain
(19)P(λ)=λ3+λ2(u*+v*u*+w*)+λ(v*+u*w*+v*w*u*+β2v*w*+k(v*)2u*)+v*w*+β2u*v*w*+k(v*)2w*u*,
and it is easy to see that Routh-Hurwitz criterion yields stability of EE.Corollary 4.
(I) Ifβ>dk+d/s, then EE exists and is locally asymptotically stable.
(II) Ifβ<dk+d/s, then EE does not exist and DFE is locally asymptotically stable.In the original model parameters Condition (I) isβr>dk+r2d/s, which means that EE exists and is stable when the disease spreads with sufficiently large coefficient β, but the reproduction rate r of predators cannot be large at the same time.


### 3.2.  Global Stability

First, we find conditions for global stability of DFE in the setℰ.Theorem 5.
Ifβ<kd+d/s, k<2, and 1-ks>δ≥s/(2-k), then DFE is globally stable in ℰ.Proof.
We define the Lyapunov function(20)L1(u,v,w)=u-u¯-u¯ln⁡⁡uu¯+v-v¯-v¯ln⁡⁡vv¯+w,
and calculating the derivative of L1 along the solution of (9) we get
(21)L1′(u,v,w)=(u-u¯)(1-u-kv)+(v-v¯)×(s-vu-βw)+βvw-dw-w2≤-(u-u¯)2-k(u-u¯)(v-v¯)+(v-v¯)uv¯-u¯vuu¯+(βv¯-d)w-w2.
Due to the assumption on β we have βv¯-d<0, and therefore
(22)L1′(u,v,w)≤-(u-u¯)2-k(u-u¯)(v-v¯)+(v-v¯)uv¯-u¯vuu¯-w2≤-((1-k2-s2u)(u-u¯)2+(1u-k2-s2u)(v-v¯)2((1-k2-s2u)(u-u¯)2).
In ℰ we have u∈[δ,1], and hence
(23)1u-k2-s2u≥1-k2-s2u≥1-k2-s2δ≥0,
due to the assumptions on k, s, and δ. This completes the proof.Notice that ford→0 the first inequality assumed in Theorem 5 is not satisfied. This means that if the death rate of infected predators is small, then the range of stability of DFE is small as well. Moreover, as k<2, to have 1-ks>0, one needs s≤0.5; for example, if k=1, then s<0.5 and 1-s>δ≥s, so δ=0.5 is a good choice independently of s.Now, we turn to the problem of global stability of EE inℰ.Theorem 6.
Ifβ>kd+d/s, k<2, and 1-ks>δ≥(1-u*)/(k(2-k)u*), then EE is globally stable in ℰ.Proof.
We define the Lyapunov function(24)L2(u,v,w)=u-u*-u*ln⁡⁡uu*+v-v*-v*ln⁡⁡vv*+w-w*-w*ln⁡⁡ww*,
and calculating the derivative of L2 along the solution of (9) we get
(25)L2′(u,v,w)=(u-u*)(1-u-kv)+(v-v*)×(s-vu-βw)+(w-w*)(βv-d-w)=-(u-u*)2-k(u-u*)(v-v*)+(v-v*)uv*-u*vuu*-(w-w*)(w-w*)=-(u-u*)2-1u(v-v*)2-(w-w*)2+(v*uu*-k)(u-u*)(v-v*)≤-((1-v*2uu*-k2)(u-u*)2+(1u-v*2uu*-k2)(v-v*)2+(w-w*)((1-v*2uu*-k2)(u-u*)2).
As in the proof of Theorem 5 we easily check that
(26)1u-v*2uu*-k2≥1-v*2uu*-k2≥0
due to the assumptions. Thus the proof is completed.Notice that asd→0 existence of the state EE is guaranteed independently of the values of k and s. As before, the inequalities k<2 and s≤0.5 hold. One can check that if d→0, then the assumptions of Theorem 6 become β>0 and the others are the same as in Theorem 5 meaning that there is large set of parameter values for which EE is globally stable.


## 3.1. Local Stability of DFE and EE

Jacobi matrix for (9) reads
(16)J(u,v,w)=(1-2u2-kv-ku0v2u2s-2vu-βw-βv0βwβv-d-2w).
For DFE we have the following relations: 1-u¯-kv¯=0, v¯=su¯, and hence
(17)JDFE=(-u¯-ku¯0s2-s-βv¯00βv¯-d),
implying that local stability of DFE depends on the sign of βv¯-d, as submatrix (-u¯-ku¯s2-s) generates eigenvalues with negative real parts. We easily see that βv¯-d>0 is equivalent to β>dk+(d/s), that is, to the existence of EE, according to Remark 1.For EE we have the relations1-u*-kv*=0, s-(v*/u*)-βw*=0, and βv*-d-w*=0, and therefore
(18)JEE=(-u*-ku*0(v*u*)2-v*u*-βv*0βw*-w*).
Calculating characteristic polynomial for EE we obtain
(19)P(λ)=λ3+λ2(u*+v*u*+w*)+λ(v*+u*w*+v*w*u*+β2v*w*+k(v*)2u*)+v*w*+β2u*v*w*+k(v*)2w*u*,
and it is easy to see that Routh-Hurwitz criterion yields stability of EE.Corollary 4.
(I) Ifβ>dk+d/s, then EE exists and is locally asymptotically stable.
(II) Ifβ<dk+d/s, then EE does not exist and DFE is locally asymptotically stable.In the original model parameters Condition (I) isβr>dk+r2d/s, which means that EE exists and is stable when the disease spreads with sufficiently large coefficient β, but the reproduction rate r of predators cannot be large at the same time.


## 3.2.  Global Stability

First, we find conditions for global stability of DFE in the setℰ.Theorem 5.
Ifβ<kd+d/s, k<2, and 1-ks>δ≥s/(2-k), then DFE is globally stable in ℰ.Proof.
We define the Lyapunov function(20)L1(u,v,w)=u-u¯-u¯ln⁡⁡uu¯+v-v¯-v¯ln⁡⁡vv¯+w,
and calculating the derivative of L1 along the solution of (9) we get
(21)L1′(u,v,w)=(u-u¯)(1-u-kv)+(v-v¯)×(s-vu-βw)+βvw-dw-w2≤-(u-u¯)2-k(u-u¯)(v-v¯)+(v-v¯)uv¯-u¯vuu¯+(βv¯-d)w-w2.
Due to the assumption on β we have βv¯-d<0, and therefore
(22)L1′(u,v,w)≤-(u-u¯)2-k(u-u¯)(v-v¯)+(v-v¯)uv¯-u¯vuu¯-w2≤-((1-k2-s2u)(u-u¯)2+(1u-k2-s2u)(v-v¯)2((1-k2-s2u)(u-u¯)2).
In ℰ we have u∈[δ,1], and hence
(23)1u-k2-s2u≥1-k2-s2u≥1-k2-s2δ≥0,
due to the assumptions on k, s, and δ. This completes the proof.Notice that ford→0 the first inequality assumed in Theorem 5 is not satisfied. This means that if the death rate of infected predators is small, then the range of stability of DFE is small as well. Moreover, as k<2, to have 1-ks>0, one needs s≤0.5; for example, if k=1, then s<0.5 and 1-s>δ≥s, so δ=0.5 is a good choice independently of s.Now, we turn to the problem of global stability of EE inℰ.Theorem 6.
Ifβ>kd+d/s, k<2, and 1-ks>δ≥(1-u*)/(k(2-k)u*), then EE is globally stable in ℰ.Proof.
We define the Lyapunov function(24)L2(u,v,w)=u-u*-u*ln⁡⁡uu*+v-v*-v*ln⁡⁡vv*+w-w*-w*ln⁡⁡ww*,
and calculating the derivative of L2 along the solution of (9) we get
(25)L2′(u,v,w)=(u-u*)(1-u-kv)+(v-v*)×(s-vu-βw)+(w-w*)(βv-d-w)=-(u-u*)2-k(u-u*)(v-v*)+(v-v*)uv*-u*vuu*-(w-w*)(w-w*)=-(u-u*)2-1u(v-v*)2-(w-w*)2+(v*uu*-k)(u-u*)(v-v*)≤-((1-v*2uu*-k2)(u-u*)2+(1u-v*2uu*-k2)(v-v*)2+(w-w*)((1-v*2uu*-k2)(u-u*)2).
As in the proof of Theorem 5 we easily check that
(26)1u-v*2uu*-k2≥1-v*2uu*-k2≥0
due to the assumptions. Thus the proof is completed.Notice that asd→0 existence of the state EE is guaranteed independently of the values of k and s. As before, the inequalities k<2 and s≤0.5 hold. One can check that if d→0, then the assumptions of Theorem 6 become β>0 and the others are the same as in Theorem 5 meaning that there is large set of parameter values for which EE is globally stable.


## 4. Analysis of (6)

We are looking for classical solutions of (6), and therefore we need to specify the proper space of initial conditions; cf. for example, [31]. Let C2+α denote the space of twice differentiable functions with Hölder coefficient α∈(0,1) and let C1+α/2,2+α denote the space C1+α/2 with respect to the time variable t and C2+α with respect to the space variable x.Proposition 7.
LetV(0,·)=(u(0,·),v(0,·),w(0,·))∈C2+α(Ω¯) and u(0,·)>0, v(0,·)≥0, w(0,·)≥0. Then there exists unique solution of (6).Proof.
Letf(V) denote the vector of kinetics for (6). The function f is locally Lipschitz continuous, because the coordinates fi, i=1,2,3, are either polynomials of the second degree or rational functions well defined for u>0. Hence, for V(0,·)∈C2+α, there exists Tmax⁡>0 such that, for every 0<T<Tmax⁡, there is a unique local solution V∈C1+α/2,2+α(Ω¯×[0,T]) of (6) (compare, e.g., [31]).Next, we would like to show nonnegativity and global existence of solutions.


### 4.1. Invariant Sets

In this subsection, we use the framework of invariant sets to prove global existence of solutions; compare [32, 33]. We show that the set
(27)𝒟=[0,1]×[0,s]×[0,βs-d],βs>d,
is invariant for (6), as in the case without diffusion, that is, for (9). As before, if βs≤d we can substitute the last interval [0,βs-d] with [0,βs].Following the ideas presented in [32] we look for the functions G:ℝ3→ℝ such that G(X)≤0 in 𝒟, G is quasi-convex, ∇G is the left eigenvector for the diffusion coefficients matrix D (D=diag⁡(di)i=13 in our case), and ∇Gf|∂𝒟≤0, where f denotes the right-hand side kinetic function of the studied system.In the case of (6) studied in this section the matrix D is diagonal, and hence every vector is an eigenvector for this matrix. Therefore, we can use functions Gi, i=1, 2, 3, which are linear as functions of appropriate variable u, v, w, respectively.To show nonnegativity we use the functions(28)G1(V)=-u≤0,G2(V)=-v≤0,G3(V)=-w≤0,forV∈𝒟.
We have
(29)∇G1(V)f(V)=-u(1-u-kv)≤0,∇G2(V)f(V)=-v(s-vu-βw)≤0,∇G3(V)f(V)=-w(βv-d-w)≤0on⁡∂𝒟.
Next we use
(30)G1(V)=u-uM≤0,G2(V)=v-vM≤0,G3(V)=w-wM≤0,forV∈𝒟,
where uM=1, vM=s, and wM=βs-d, βs>d (or wM=βs, βs≤d). For these functions we obtain
(31)∇G1(V)f(V)=u(1-u-kv)≤u(1-u)|u=uM=0,∇G2(V)f(V)=v(s-vu-βw)≤v(s-vuM)|v=vM=0,∇G3(V)f(V)=w(βv-d-w)≤w(βvM-d-w)|w=wM=0.
Therefore, 𝒟 is invariant according to the theory of invariant sets for RDEs.The theory of invariant sets implies that, as in the case described above, if there exists a compact invariant set, then solutions of the studied system and initial data from this invariant set are global in time. This leads to the global existence of nonnegative solutions of (6) for nonnegative initial data.In the next section, we will investigate long-time behavior of (6), including existence of global attractor and persistence property.


## 4.1. Invariant Sets

In this subsection, we use the framework of invariant sets to prove global existence of solutions; compare [32, 33]. We show that the set
(27)𝒟=[0,1]×[0,s]×[0,βs-d],βs>d,
is invariant for (6), as in the case without diffusion, that is, for (9). As before, if βs≤d we can substitute the last interval [0,βs-d] with [0,βs].Following the ideas presented in [32] we look for the functions G:ℝ3→ℝ such that G(X)≤0 in 𝒟, G is quasi-convex, ∇G is the left eigenvector for the diffusion coefficients matrix D (D=diag⁡(di)i=13 in our case), and ∇Gf|∂𝒟≤0, where f denotes the right-hand side kinetic function of the studied system.In the case of (6) studied in this section the matrix D is diagonal, and hence every vector is an eigenvector for this matrix. Therefore, we can use functions Gi, i=1, 2, 3, which are linear as functions of appropriate variable u, v, w, respectively.To show nonnegativity we use the functions(28)G1(V)=-u≤0,G2(V)=-v≤0,G3(V)=-w≤0,forV∈𝒟.
We have
(29)∇G1(V)f(V)=-u(1-u-kv)≤0,∇G2(V)f(V)=-v(s-vu-βw)≤0,∇G3(V)f(V)=-w(βv-d-w)≤0on⁡∂𝒟.
Next we use
(30)G1(V)=u-uM≤0,G2(V)=v-vM≤0,G3(V)=w-wM≤0,forV∈𝒟,
where uM=1, vM=s, and wM=βs-d, βs>d (or wM=βs, βs≤d). For these functions we obtain
(31)∇G1(V)f(V)=u(1-u-kv)≤u(1-u)|u=uM=0,∇G2(V)f(V)=v(s-vu-βw)≤v(s-vuM)|v=vM=0,∇G3(V)f(V)=w(βv-d-w)≤w(βvM-d-w)|w=wM=0.
Therefore, 𝒟 is invariant according to the theory of invariant sets for RDEs.The theory of invariant sets implies that, as in the case described above, if there exists a compact invariant set, then solutions of the studied system and initial data from this invariant set are global in time. This leads to the global existence of nonnegative solutions of (6) for nonnegative initial data.In the next section, we will investigate long-time behavior of (6), including existence of global attractor and persistence property.


## 5. Long-Time Behavior of Solutions of (6)

First we focus on the persistence of the system described by (6) which is closely related to the existence of the positive equilibrium EE. We use the comparison principle for parabolic systems (cf., e.g., [34]) to show desired properties of solution of (6). As a system for comparison, we take the logistic equation with diffusion and zero-flux boundary conditions.


### 5.1. Global Attractor and Persistence Property

Lemma 8 (see [35]).
Assume thatu(x,t) is a solution of the problem
(32)∂u∂t=d1Δu+ru(1-uK),x∈Ω,t>0,∂u∂η=0,x∈∂Ω,t>0,u(x,0)=u0(x)>0,x∈Ω;
then lim⁡t→∞u(x,t)=K.Lemma 9.
Solutions of (6) satisfy
(33)lim⁡⁡sup⁡⁡t→∞max⁡⁡x∈Ω¯u(x,t)≤1,lim⁡⁡sup⁡⁡t→∞max⁡⁡x∈Ω¯v(x,t)≤s,lim⁡⁡sup⁡⁡t→∞max⁡⁡x∈Ω¯w(x,t)≤βs-d,forβs>d,orlim⁡⁡sup⁡⁡t→∞max⁡⁡x∈Ω¯w(·,t)≤βs,forβs≤d.Proof.
Due to nonnegativity of solutions, from the first equation of (6), we have
(34)∂u∂t≤d1Δu+u(1-u),
and we compare the solution of our problem with the solution of (32). Therefore, for an arbitrary ε>0 there exists T∈(0,∞) such that u(x,t)≤1+ε for (x,t)∈Ω¯×[T,∞).
Next, from the second equation we have(35)∂v∂t≤d2Δv+v(s-v1+ε)⟹lim⁡⁡sup⁡⁡t→∞max⁡⁡x∈Ω¯v(x,t)≤s(1+ε),
and for any ε1>0 there exists T1∈(T,∞) such that v(x,t)≤s(1+ε1) in Ω¯×[T1,∞).
Finally, from the third equation, we have(36)∂w∂t≤d3Δw+w(β(s+sε1)-d-w),
and hence
(37)lim⁡⁡sup⁡⁡t→∞max⁡⁡x∈Ω¯w(x,t)≤β(s+sε1)-d(or≤β(s+sε1)),
and the proof is completed, as ε and ε1 are arbitrary.Theorem 10.
Ifks<1, βs>d, and β(s(1-β2)+βd)(1-ks)>d, then the system described by (6) is persistent.Proof.
Below we use upper bounds onu, v, and w obtained in Lemma 9. From the first equation of (6), we have
(38)∂u∂t≥d1Δu+u(1-u)-kus(1+ε)=u(1-ks(1+ε)-u),
for ε>0 such that ks(1+ε)<1. Then, by the comparison principle and Lemma 8, we easily get u(x,t)≥1-ks(1+ε) for t large enough. As ε is arbitrary, we obtain
(39)lim⁡⁡inf⁡⁡t→∞min⁡⁡x∈Ω¯u(x,t)≥1-ks≐m1.
Next, for anyε2>0 and t large enough, from the second equation, we have
(40)∂v∂t≥d2Δv+v(s-v1-ks(1+ε)-β(βs-d+ε2)),
and hence
(41)lim⁡⁡inf⁡⁡t→∞max⁡⁡x∈Ω¯v(x,t)≥(s-β(βs-d))(1-ks)≐m2.
From the third equation of (6), we have
(42)∂w∂t≥d3Δw+w(β(m2+ε3)-d-w),
for arbitrary small ε3>0 and t large enough. Again by the comparison principle and Lemma 8, we have
(43)lim⁡⁡inf⁡⁡t→∞min⁡⁡x∈Ω¯w(x,t)≥βm2-d≐m3.
Taking into account Lemma 9 we easily obtain the persistence property.


### 5.2. Local and Global Stability of the Positive Equilibrium EE

First, we discuss local stability of the positive equilibrium EE under the influence of diffusion.Proposition 11.
Ifβ>dk+d/s, then the unique positive equilibrium EE of (6) exists and is locally asymptotically stable.Proof.
Linearizing (6) around EE we obtain
(44)Vt=(DΔ+JEE)V,
where V=(u(x,t), v(x,t), w(x,t))T, D=diag⁡(d1,d2,d3), and JEE is the Jacobi matrix for (9) evaluated at the point (u*,v*,w*). Assuming for simplicity that Ω=(0,π), we have fundamental solutions of the form eλtcos⁡⁡(μix), i∈ℕ. Therefore, for any wave number i the matrix JEE changes to
(45)JEEi=(-d1μi2-u*-kv*0v*2u*2-d2μi2-v*u*-βv*0βw*-d3μi2-w*),
with the characteristic polynomial
(46)Pi(λ)=λ3+λ2(u*+d1μi2+v*u*+d2μi2+w*+d3μi2)+λ(+β2v*w*+k(v*)2u*)(u*+d1μi2)(v*u*+d2μi2)+(u*+d1μi2)(w*+d3μi2)+(v*u*+d2μi2)(w*+d3μi2)+β2v*w*+k(v*)2u*)+(u*+d1μi2)(v*u*+d2μi2)(w*+d3μi2)+β2u*v*w*+k(v*)2w*u*,
and it is again easy to see that Routh-Hurwitz criterion yields stability of EE independently of the magnitude of di, i=1, 2, 3.In the following, we focus on global stability of EE, which implies that the three populations coexisting in the ecosystem will be spatially homogeneously distributed with increasing time.Theorem 12.
Ifβ>dk+d/s and 1-(v*/2m1u*)-(k/2)>0, m1=1-ks>0, then the unique positive equilibrium EE of (6) is globally asymptotically stable.Proof.
We construct a Lyapunov functional on the basis of the functionL2 for (9). Let us define Z(t)=∫ΩL2(u(x,t),v(x,t),w(x,t))dx. Differentiating Z along the trajectories of (6) we obtain
(47)dZ(t)dt=∫Ω(∂L2∂uut+∂L2∂vvt+∂L2∂wwt)dx=∫Ω(u-u*ud1Δu+v-v*vd2Δv+w-w*wd3Δw)dx+∫Ω((sb-vu-βw)(u-u*)(1-u-kv)+(v-v*)(sb-vu-βw)+(w-w*)(βv-d-w)(sb-vu-βw))dx,
and due to the Neumann boundary condition, we have
(48)dE(t)dt=-∫Ω(d1u*u2|∇u|2+d2v*v2|∇v|2+d3w*w2|∇w|2)dx-∫Ω((u-u*)2+1u(v-v*)2+(w-w*)2)dx+∫Ω((v*uu*-k)(u-u*)(v-v*))dx≤-∫Ω(d1u*u2|∇u|2+d2v*v2|∇v|2+d3w*w2|∇w|2)dx-∫Ω((w-w*)2+(1-v*2m1u*-k2)(u-u*)2+(1-v*2m1u*-k2)(v-v*)2)dx≤0
which implies the desired assertion.


### 5.3. Local and Global Stability of Semitrivial DFE Equilibrium

Using the same approach as for the positive equilibrium we can easily show that diffusion has no influence on the dynamics of (6) also when the positive equilibrium does not exist.Corollary 13.
Ifβ<dk+d/s and 1-k/2-s/2m1>0, m1=1-ks>0, then the semitrivial equilibrium DFE of (6) is globally asymptotically stable.


## 5.1. Global Attractor and Persistence Property

Lemma 8 (see [35]).
Assume thatu(x,t) is a solution of the problem
(32)∂u∂t=d1Δu+ru(1-uK),x∈Ω,t>0,∂u∂η=0,x∈∂Ω,t>0,u(x,0)=u0(x)>0,x∈Ω;
then lim⁡t→∞u(x,t)=K.Lemma 9.
Solutions of (6) satisfy
(33)lim⁡⁡sup⁡⁡t→∞max⁡⁡x∈Ω¯u(x,t)≤1,lim⁡⁡sup⁡⁡t→∞max⁡⁡x∈Ω¯v(x,t)≤s,lim⁡⁡sup⁡⁡t→∞max⁡⁡x∈Ω¯w(x,t)≤βs-d,forβs>d,orlim⁡⁡sup⁡⁡t→∞max⁡⁡x∈Ω¯w(·,t)≤βs,forβs≤d.Proof.
Due to nonnegativity of solutions, from the first equation of (6), we have
(34)∂u∂t≤d1Δu+u(1-u),
and we compare the solution of our problem with the solution of (32). Therefore, for an arbitrary ε>0 there exists T∈(0,∞) such that u(x,t)≤1+ε for (x,t)∈Ω¯×[T,∞).
Next, from the second equation we have(35)∂v∂t≤d2Δv+v(s-v1+ε)⟹lim⁡⁡sup⁡⁡t→∞max⁡⁡x∈Ω¯v(x,t)≤s(1+ε),
and for any ε1>0 there exists T1∈(T,∞) such that v(x,t)≤s(1+ε1) in Ω¯×[T1,∞).
Finally, from the third equation, we have(36)∂w∂t≤d3Δw+w(β(s+sε1)-d-w),
and hence
(37)lim⁡⁡sup⁡⁡t→∞max⁡⁡x∈Ω¯w(x,t)≤β(s+sε1)-d(or≤β(s+sε1)),
and the proof is completed, as ε and ε1 are arbitrary.Theorem 10.
Ifks<1, βs>d, and β(s(1-β2)+βd)(1-ks)>d, then the system described by (6) is persistent.Proof.
Below we use upper bounds onu, v, and w obtained in Lemma 9. From the first equation of (6), we have
(38)∂u∂t≥d1Δu+u(1-u)-kus(1+ε)=u(1-ks(1+ε)-u),
for ε>0 such that ks(1+ε)<1. Then, by the comparison principle and Lemma 8, we easily get u(x,t)≥1-ks(1+ε) for t large enough. As ε is arbitrary, we obtain
(39)lim⁡⁡inf⁡⁡t→∞min⁡⁡x∈Ω¯u(x,t)≥1-ks≐m1.
Next, for anyε2>0 and t large enough, from the second equation, we have
(40)∂v∂t≥d2Δv+v(s-v1-ks(1+ε)-β(βs-d+ε2)),
and hence
(41)lim⁡⁡inf⁡⁡t→∞max⁡⁡x∈Ω¯v(x,t)≥(s-β(βs-d))(1-ks)≐m2.
From the third equation of (6), we have
(42)∂w∂t≥d3Δw+w(β(m2+ε3)-d-w),
for arbitrary small ε3>0 and t large enough. Again by the comparison principle and Lemma 8, we have
(43)lim⁡⁡inf⁡⁡t→∞min⁡⁡x∈Ω¯w(x,t)≥βm2-d≐m3.
Taking into account Lemma 9 we easily obtain the persistence property.


## 5.2. Local and Global Stability of the Positive Equilibrium EE

First, we discuss local stability of the positive equilibrium EE under the influence of diffusion.Proposition 11.
Ifβ>dk+d/s, then the unique positive equilibrium EE of (6) exists and is locally asymptotically stable.Proof.
Linearizing (6) around EE we obtain
(44)Vt=(DΔ+JEE)V,
where V=(u(x,t), v(x,t), w(x,t))T, D=diag⁡(d1,d2,d3), and JEE is the Jacobi matrix for (9) evaluated at the point (u*,v*,w*). Assuming for simplicity that Ω=(0,π), we have fundamental solutions of the form eλtcos⁡⁡(μix), i∈ℕ. Therefore, for any wave number i the matrix JEE changes to
(45)JEEi=(-d1μi2-u*-kv*0v*2u*2-d2μi2-v*u*-βv*0βw*-d3μi2-w*),
with the characteristic polynomial
(46)Pi(λ)=λ3+λ2(u*+d1μi2+v*u*+d2μi2+w*+d3μi2)+λ(+β2v*w*+k(v*)2u*)(u*+d1μi2)(v*u*+d2μi2)+(u*+d1μi2)(w*+d3μi2)+(v*u*+d2μi2)(w*+d3μi2)+β2v*w*+k(v*)2u*)+(u*+d1μi2)(v*u*+d2μi2)(w*+d3μi2)+β2u*v*w*+k(v*)2w*u*,
and it is again easy to see that Routh-Hurwitz criterion yields stability of EE independently of the magnitude of di, i=1, 2, 3.In the following, we focus on global stability of EE, which implies that the three populations coexisting in the ecosystem will be spatially homogeneously distributed with increasing time.Theorem 12.
Ifβ>dk+d/s and 1-(v*/2m1u*)-(k/2)>0, m1=1-ks>0, then the unique positive equilibrium EE of (6) is globally asymptotically stable.Proof.
We construct a Lyapunov functional on the basis of the functionL2 for (9). Let us define Z(t)=∫ΩL2(u(x,t),v(x,t),w(x,t))dx. Differentiating Z along the trajectories of (6) we obtain
(47)dZ(t)dt=∫Ω(∂L2∂uut+∂L2∂vvt+∂L2∂wwt)dx=∫Ω(u-u*ud1Δu+v-v*vd2Δv+w-w*wd3Δw)dx+∫Ω((sb-vu-βw)(u-u*)(1-u-kv)+(v-v*)(sb-vu-βw)+(w-w*)(βv-d-w)(sb-vu-βw))dx,
and due to the Neumann boundary condition, we have
(48)dE(t)dt=-∫Ω(d1u*u2|∇u|2+d2v*v2|∇v|2+d3w*w2|∇w|2)dx-∫Ω((u-u*)2+1u(v-v*)2+(w-w*)2)dx+∫Ω((v*uu*-k)(u-u*)(v-v*))dx≤-∫Ω(d1u*u2|∇u|2+d2v*v2|∇v|2+d3w*w2|∇w|2)dx-∫Ω((w-w*)2+(1-v*2m1u*-k2)(u-u*)2+(1-v*2m1u*-k2)(v-v*)2)dx≤0
which implies the desired assertion.


## 5.3. Local and Global Stability of Semitrivial DFE Equilibrium

Using the same approach as for the positive equilibrium we can easily show that diffusion has no influence on the dynamics of (6) also when the positive equilibrium does not exist.Corollary 13.
Ifβ<dk+d/s and 1-k/2-s/2m1>0, m1=1-ks>0, then the semitrivial equilibrium DFE of (6) is globally asymptotically stable.


## 6. Discussion

In the paper we have considered a prey-predator ecosystem in which the predator species is infected, such that three species (prey, healthy predators, and infected predators) are described in ecoepidemiological model based on the May predator-prey model from ecological point of view and SI model from epidemiological point of view. We have studied the influence of spatial effects incorporating simple diffusion equipped with zero-flux boundary condition into the model. Our studies included global existence and uniqueness of solutions, which is not a common procedure in the papers devoted to biological modeling but is important and can be a nontrivial, difficult issue (cf., e.g., [36]).We have shown that the system dynamics mainly depends on the infection rate parameterβ. When β is small, there is no positive equilibrium and we expect that the population of infected predators will become extinct. This is exactly the result we obtained. We have formulated conditions for local and global stability of the semitrivial equilibrium which reflects the case of ecosystem without the disease. For β large enough the positive equilibrium describing the disease endemic state exists and it can be expected that it is stable. In fact, there is a bifurcation at some threshold value βth, such that for β<βth there is no positive equilibrium and the semitrivial equilibrium is locally asymptotically stable, while for β above this threshold, the semitrivial equilibrium loses stability and the positive stable equilibrium exists and is stable. We have also obtained conditions on global stability of the positive equilibrium.Our main result shows that the diffusion has no significant influence on the model dynamics; that is, local and global stability do not depend on the magnitude of diffusion coefficients. We suspect that such type of the model dynamics can be a result of simple form of diffusion incorporated into the model, as other types of diffusion can lead to more complex dynamics; cf., for example, ([34–37]). However, in our opinion, without complex studies on specific species it is difficult to recognize the law governing the process of diffusion, and therefore simple diffusion seems to be a good approximation.



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*Source: 236208-2014-04-29.xml*