Asymptotic behavior of a diffusive eco-epidemiological model with an infected prey population

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Abstract

We study a diffusive predator-prey system with a ratio-dependent functional response when a prey population is infected under homogeneous Neumann boundary condition. All non-negative and positive equilibria are investigated, and the conditions that give rise to asymptotic behavior of these equilibria are examined. In particular, we present a biological interpretation of disease-free and total extinction states. A comparison principle and the stability analysis for the parabolic problem are employed.

Introduction

We focus on the diffusive predator-prey system with a ratio-dependent functional response and disease in the prey; specifically,

$$\textstyle\begin{cases} u_{t} -d \Delta u = u [r-\frac{r}{K}u-\frac{\alpha w}{mw+u+v}-bv], \\ v_{t} -d \Delta v = v [bu-d_{1} -\frac{\beta w}{mw+u+v}], \\ w_{t} -D \Delta w = w [-d_{2} +\frac{c\alpha u}{mw+u+v}+\frac{c\beta v}{mw+u+v}] \quad \mbox{in (0,\infty)\times\Omega,} \\ \frac{\partial u}{\partial\eta}=\frac{\partial v}{\partial\eta}=\frac{\partial w}{\partial\eta}=0 \quad \mbox{on (0,\infty)\times\partial \Omega,} \\ u(0,x)=u_{0} (x),\qquad v(0,x)=v_{0} (x),\qquad w(0,x)=w_{0} (x) \quad \mbox{in \Omega,} \end{cases}$$
(1.1)

where $$\Omega\subseteq\mathbb{R}^{N}$$ is a bounded region with smooth boundary Ω, and r, m, K, b, $$d_{i}$$, $$D_{i}$$, c, α, and β are positive constants; $$a, b_{1}, b_{2}, l$$ and k are positive constants as well. The initial functions $${u}_{0}$$, $${v}_{0}$$, and $${w}_{0}$$ are not identically zero in Ω; u, v, and w represent the densities of the susceptible prey, infected prey, and predator, respectively, and η is the outward directional derivative normal to Ω. Furthermore, α and β are the searching efficiency constants of the predation rate for the susceptible and infective prey, respectively. $$\frac{\alpha}{m}$$ and $$\frac{\beta}{m}$$ are the maximum per capita capturing rates of the predator for the susceptible prey and infected prey, respectively. m is the predation rate for the susceptible prey and infected prey. Finally, b is the force of infection, $$d_{1}$$ and $$d_{2}$$ are the death rates of the infected prey and predator, respectively, and c is a conversion rate. The homogeneous Neumann boundary condition describes an environment with no flux at the boundary of the region.

During the last three decades, various types of predator-prey models are studied extensively by many researchers. Many models have a functional response in which ignoring an effect of predator density, i.e., the function that describe a density of prey which is consumed by its predator depends only on prey. However, there is explicit biological and physiological evidence  that in many situations, when predator have to search for food, a more suitable general predator-prey model in heterogeneous situations should be had the ratio-dependent functional response, which the per capita predator growth rate should be a function of the ratio of prey to predator abundance. Ratio-dependent models have been mathematically studied for both the spatially homogeneous case  and the spatially inhomogeneous case . In [8, 12], one examined the model of Arditi and Ginzburg . One showed that under some conditions, the whole population can be extinct.

On the other hand, epidemic models have also received a lot of attention since Kermack-McKendrick’s model. Among them, we are interested in eco-epidemiological systems with predator-prey interactions. Considerable research has been done on the spatially homogeneous case .

In the real application, diffusive system in this study can be used to describe the interaction between marine viruses in aquatic ecosystems and the species [21, 22], since there is evidence that viral infection might accelerate the termination of phytoplankton blooms . In fact, in , the authors showed experimentally that viral disease can infect bacteria and phytoplankton in coastal water. In [14, 25, 26], they observed oscillations and waves in a phytoplankton-zooplankton system with Holling-type II and III grazing under lysogenic viral infection and frequency-dependent transmission. Hilker  also investigated the local dynamics of phytoplankton with lytic infection and frequency-dependent transmission as well as zooplankton with Holling-type II grazing.

Arino et al.  suggested the non-dimensionalized model, which is a non-spatial version of (1.1). There, the authors obtained the conditions for which no trajectory can reach the origin following any fixed direction or spirally. Also the criteria of persistence were found. The above studies have been done mostly for the non-spatial case.

In this paper, we investigate the conditions of the asymptotic behavior of a unique positive constant solution and the non-negative equilibria of (1.1), which is a spatially dependent model with diffusion.

Model (1.1) is based on the following assumptions:

1. (a)

In the absence of disease, the prey population grows according to logistic law with carrying capacity $$K > 0$$ and an intrinsic growth rate $$r >0$$.

2. (b)

In the presence of disease, the prey consists of two classes: susceptible prey and infected prey.

3. (c)

Only susceptible prey can reproduce themselves logistically and contribute to its carrying capacity. Infected prey do not grow, recover, or reproduce.

4. (d)

Disease can only be spread among the prey, and it is not inherited. Disease transmission follows the simple law of mass action.

From the literature , the assumption (c) can be justified in many cases: the experiment on dinoflagellate Noctiluca scintillans in the German Bight by Uhlig and Sahling  indicated that the cells become damaged, and they neither feed anymore nor reproduce. The model of Hamilton et al.  showed that infected individuals do not contribute in the reproduction process; infection reduces the remaining capacity due to the inability to compete for resources. Thus, we may assume that the growth term of the susceptible population follows only the law of logistic growth.

For additional background information pertaining to (1.1), we refer to  and the references therein.

The remainder of this paper is organized as follows. In Section 2, we investigate the large time behavior of non-negative constant solutions and the asymptotic stability of a positive constant solution. Finally, the results obtained are analyzed in terms of biological interpretations in Section 3.

Asymptotical behavior of constant solutions

In this section, the asymptotic behavior of non-negative and positive constant solutions to (1.1) is examined.

For convenience, we denote the growth rate terms as follows:

\begin{aligned} &f_{1} (u,v,w) := r-\frac{r}{K}u-\frac{\alpha w}{mw+u+v}-bv, \\ &f_{2} (u,v,w) :=bu-d_{1} -\frac{\beta w}{mw+u+v}, \\ &f_{3} (u,v,w) := -d_{2} +\frac{c\alpha u}{mw+u+v}+ \frac{c\beta v}{mw+u+v}. \end{aligned}

Using the uniform bound of u, v and w, one can show that $$(uf_{1}, vf_{2}, wf_{3} )$$ satisfies the Lipschitz condition. Using the upper and lower solution method in , it can also be shown that (1.1) has a non-negative solution.

The next theorem states that the solution to (1.1) is uniformly bounded .

Theorem 2.1

The solution $$(u,v,w)$$ of (1.1) is uniformly bounded; specifically,

$$0 \leq u(t,x) \leq B_{1},\qquad 0 \leq v(t,x) \leq B_{2}, \qquad 0 \leq w(t,x) \leq B_{3},$$

where $$B_{i}$$ is defined by

\begin{aligned} &B_{1} := \max \bigl\{ K, \Vert u_{0} \Vert _{\infty} \bigr\} , \\ &B_{2} := \max \biggl\{ \frac{1}{d_{1}} \frac{K}{r} \biggl( \frac{r+d_{1}}{2} \biggr)^{2} , \Vert u_{0} \Vert _{\infty} + \Vert v_{0} \Vert _{\infty} \biggr\} , \\ &B_{3} := \max \biggl\{ \Vert w_{0} \Vert _{\infty}, \frac{c(\alpha +\beta)-d_{2}}{d_{2} m} B_{2} \biggr\} . \end{aligned}

The dissipation and persistence of the parabolic system (1.1) can be found in .

Theorem 2.2

For a solution $$\mathbf{u} =(u(t,x), v(t,x), w(t,x))$$ to the parabolic system (1.1),

$$\limsup_{t\rightarrow\infty} \mathbf{u} \leq \biggl(K, \frac{1}{d_{1}} \frac{K}{r} \biggl(\frac{r+d_{1}}{2} \biggr)^{2}, \frac{c(\alpha+\beta)-d_{2}}{d_{2} m} \frac{1}{d_{1}}\frac{K}{r} \biggl(\frac{r+d_{1}}{2} \biggr)^{2} \biggr)$$

if $$c(\alpha+\beta)>d_{2}$$.

Theorem 2.3

Assume that $$\beta\geq\alpha>\frac{d_{2}}{c}$$, $$r> \min\{\frac{b}{d_{1}}\frac{K}{r} (\frac{r+d_{1}}{2} )^{2} +\frac{\alpha}{m}, \frac{1}{b}\frac{r}{K} (d_{1} + \frac{\beta}{m} ) +\frac{\alpha}{m}\}$$. Then

$$\liminf_{t\rightarrow\infty} \mathbf{u} \geq (\Theta_{1} , \Theta_{2}, \Theta_{3} ),$$

where $$\Theta_{1}:= (r-\frac{b}{d_{1}}\frac{K}{r} (\frac{r+d_{1}}{2} )^{2} -\frac{\alpha}{m} )\frac{K}{r}$$, $$\Theta_{2}:=\frac{1}{b} (r-\frac{1}{b}\frac{r}{K} (d_{1} + \frac{\beta}{m} ) -\frac{\alpha}{m} )$$, and $$\Theta _{3}:=\frac{c\alpha-d_{2}}{d_{2} m} \Theta_{1}$$ for $$\frac{\alpha}{m^{2} \Theta_{3}} \leq b$$.

Equilibria

System (1.1) has the following non-negative equilibria:

$$\textstyle\begin{cases} \mathbf{e_{0}} =(0,0,0),&\\ \mathbf{e_{1}} =(K,0,0),&\\ \mathbf{e_{2}} = (K(1-\frac{c\alpha-d_{2}}{cmr}), 0, \frac{c\alpha -d_{2}}{d_{2} m}K(1-\frac{c\alpha-d_{2}}{cmr}) ) \quad \mbox{if 0< c\alpha-d_{2} < cmr,} \\ \mathbf{e_{3}} =(\frac{d_{1} }{b}, \frac{r}{b}(1-\frac{d_{1}}{bK}),0) \quad \mbox{if bK>d_{1}.} \end{cases}$$
(2.1)

Note that the given growth rates in (1.1) are not defined at $$(u,v,w)=(0, 0, 0)$$. Since

$$\lim_{(u, v, w)\rightarrow(0, 0,0)} \frac{uw}{ mw + u + v}=\lim_{(u, v,w)\rightarrow(0, 0,0)} \frac{ vw}{mw + u + v}=0,$$

the domain of $$\frac{uw}{m w + u + v}$$ and $$\frac{vw}{ m w + u + v}$$ may be extended to $$\{(u,v, w): u\geq0, v\geq0, w \geq0\}$$ so that $$(0,0, 0)$$ becomes a trivial solution to (1.1) .

Furthermore, if the following conditions are satisfied:

$$AS^{2} +BS +C< 0\quad \mbox{and} \quad d_{2} < c \beta,$$
(2.2)

where

\begin{aligned} &A=cmr\beta b, \\ &B=-c \bigl[r\beta(\beta+md_{1} ) +mk(r\beta+\alpha d_{1} )b \bigr]-d_{2} \bigl(-r\beta +K b(\beta-\alpha) \bigr), \\ &C =K(r\beta+\alpha d_{1} ) \bigl[c(\beta+md_{1} )-d_{2} \bigr], \\ &S=\frac{(r\beta+\alpha d_{1} )K}{r\beta+\alpha b K}, \end{aligned}

then there exists a unique positive equilibrium point $$\mathbf{u_{*}} =(u_{*}, v_{*}, w_{*} )$$, where

\begin{aligned} &u_{*} = \frac{-B -\sqrt{B^{2} -4AC}}{2A}, \\ &v_{*} = - \biggl(\frac{r}{bK} +\frac{\alpha}{\beta} \biggr)u_{*} + \biggl( \frac{r}{b} +\frac{\alpha d_{1}}{b \beta} \biggr), \\ &w_{*} = \frac{(c\alpha-d_{2} )u_{*} +(c\beta-d_{2} )v_{*} }{d_{2} m}. \end{aligned}

Asymptotic stability of equilibria

In this subsection, we investigate the non-negative equilibria $$\mathbf{e_{0}}$$, $$\mathbf{e_{1}}$$, $$\mathbf{e_{2}}$$, and $$\mathbf{e_{3}}$$ defined in (2.1) and the positive equilibrium point $$\mathbf{u_{*}}$$.

Asymptotic stability of $$\mathbf{e_{0}}$$

We investigate the stability at $$(0,0,0)$$. For the stability of $$\mathbf{e_{0}}$$, we assume $$d=D$$.

Figure 1 shows that under some conditions, all three species become extinct. We point out that the corresponding non-spatial model had the same asymptotic behavior under the same condition in the following theorem.

Theorem 2.4

Assume that $$mc \leq1$$, $$\beta\geq\alpha$$, $$\min\{c\alpha-d_{2}, \frac{\alpha}{1+m}\} \geq r$$. If the initial data satisfies $$w_{0} \geq u_{0} +v_{0}$$, then $$\lim_{t\rightarrow\infty} \mathbf{u} =\mathbf{e_{0}}$$.

Proof

Subtracting the first and second equations from the third equation in (1.1) yields

\begin{aligned} (w-u-v)_{t} -d\Delta(w-u-v) &=wf_{3} -uf_{1} -vf_{2} \\ &=(w-u-v)f_{3} +(f_{3}-f_{2} )v +(f_{3} -f_{1} )u. \end{aligned}
(2.3)

Also note that

\begin{aligned} f_{3}-f_{2} &=\frac{c\alpha u+c\beta v +\beta w}{mw+u+v}-d_{2} -bu +d_{1} \\ &=c\alpha\frac{u+\frac{\beta}{\alpha} v +\frac{\beta}{c\alpha} w}{mw+u+v}-d_{2} -bu +d_{1} \\ &\geq c\alpha -d_{2} +d_{1} -bu, \\ f_{3} -f_{1} &=\frac{c\alpha u+c\beta v +\alpha w}{mw+u+v} -d_{2} -r + \frac{r}{K}u +bv \\ &= c\alpha\frac{u+\frac{\beta}{\alpha} v +\frac{1}{c} w}{mw+u+v} -d_{2} -r +\frac{r}{K}u +bv \\ &\geq c \alpha-d_{2} -r +\frac{r}{K}u +bv \end{aligned}

hold under the assumptions that $$mc \leq1$$ and $$\beta\geq\alpha$$. As a result, $$(f_{3}-f_{2} )v +(f_{3} -f_{1} )u \geq(c\alpha -d_{2} +d_{1} )v +(c \alpha-d_{2} -r +\frac{r}{K}u )u \geq0$$, since $$c\alpha-d_{2} \geq r$$. Thus, applying the positivity lemma  to (2.3), $$w \geq u+v$$ holds if $$w_{0} \geq u_{0} +v_{0}$$. In the light of these facts, the main result is satisfied; specifically,

\begin{aligned} u_{t} -d \Delta u &= u \biggl[r-\frac{r}{K}u- \frac{\alpha w}{mw+u+v}-bv\biggr] \\ &\leq u\biggl[r-\frac{r}{K}u-\frac{\alpha w}{mw+w}\biggr] \\ &=u\biggl[r-\frac{r}{K}u-\frac{\alpha}{m+1}\biggr] \\ &\leq0, \end{aligned}

since $$\frac{\alpha}{1+m} \geq r$$. Thus, $$\lim_{t \rightarrow\infty} w=0$$ on Ω̅. Consequently, $$\lim_{t \rightarrow\infty} u = 0$$ and $$\lim_{t \rightarrow\infty} v =0$$ on Ω̅ since $$w \geq u+v$$. □

Theorem 2.5

1. (i)

If there exists a positive constant θ such that

\begin{aligned} \begin{aligned} &rm-\alpha+d_{2} m \leq(c\alpha-r -d_{2} )\theta, \\ &d_{2} m -d_{1} m -\beta\leq(c\beta+d_{1} -d_{2} )\theta, \end{aligned} \end{aligned}
(2.4)

holds, then the region $$\Sigma=\{(u,v,w): u, v, w \geq 0, u+v \leq\theta w \}$$ is an invariant set for (1.1).

2. (ii)

In addition to (2.4), if $$-m +\frac{\alpha}{r} \geq\theta$$, $$\lim_{t\rightarrow\infty} \mathbf{u} =e_{0}$$ for the initial function $$(u_{0},v_{0},w_{0} ) \in\Sigma$$.

3. (iii)

In addition to (2.4), if $$c\max\{\alpha, \beta\} \leq d_{2}$$, $$\lim_{t\rightarrow\infty} \mathbf{u} =e_{0}$$ for the initial function $$(u_{0},v_{0},w_{0} ) \in\Sigma$$.

4. (iv)

In addition to (2.4), if $$c\beta\geq c\alpha> d_{2}$$ and $$\theta< \frac{d_{2} m}{c\alpha-d_{2} }$$, $$\lim_{t\rightarrow\infty} \mathbf{u} =e_{0}$$ for the initial function $$(u_{0},v_{0},w_{0} ) \in\Sigma$$.

Proof

(i) Let $$G(u,v,w)=u+v -\theta w$$. To achieve the desired result, Corollary 14.8 of  is used; in particular, we will show that $$(uf_{1},vf_{2},wf_{3} )$$ points into Σ on Σ. On the boundary of Σ (except for the boundary $$u+v= \theta w$$), $$dG \cdot(uf_{1},vf_{2},wf_{3} )\leq0$$ can easily be verified.

It is straightforward to show that $$dG \cdot(uf_{1},vf_{2},wf_{3} ) \leq0$$ on the boundary $$u+v= \theta w$$. In fact,

\begin{aligned} &dG \cdot(uf_{1},vf_{2},wf_{3} ) \\ &\quad = (1,1,-\theta) \cdot(uf_{1},vf_{2} ,wf_{3} ) \\ &\quad = uf_{1} +vf_{2} -\theta wf_{3} \\ &\quad=u \biggl[r-\frac{r}{K}u-\frac{\alpha w}{mw+\theta w}-bv \biggr]+v \biggl[bu-d_{1} -\frac{\beta w}{mw+\theta w} \biggr] \\ &\qquad {}-\theta w \biggl[-d_{2} +\frac{c\alpha u}{mw+\theta w}+\frac{c\beta v}{mw+\theta w} \biggr] \\ &\quad=u \biggl(r-\frac{\alpha}{m+\theta} \biggr) -\frac{r}{K}u^{2} +d_{2} \theta w -v \biggl( d_{1} +\frac{\beta}{m+\theta} \biggr)- \frac{c\alpha \theta u}{m+\theta}-\frac{c\beta\theta v}{m+\theta} \\ &\quad =u \biggl(r-\frac{\alpha}{m+\theta} +d_{2} -\frac{c\alpha \theta}{m+\theta} \biggr) +v \biggl( d_{2} -d_{1} -\frac{\beta}{m+\theta}-\frac{c \beta\theta}{m+\theta} \biggr) -\frac{r}{K}u^{2} \\ &\quad \leq0. \end{aligned}

The last inequality holds by assumption (2.4).

(ii) Since Σ is an invariant region under assumption (2.4), $$u+v\leq\theta w$$ holds for $$(u_{0},v_{0},w_{0} ) \in\Sigma$$. Thus, the following inequality is satisfied if $$u+v\leq\theta w$$ and $$-m +\frac{\alpha}{r} \geq\theta$$:

\begin{aligned} u_{t} -d \Delta u &= u \biggl[r-\frac{r}{K}u- \frac{\alpha w}{mw+u+v}-bv \biggr] \\ &\leq u \biggl[r-\frac{r}{K}u-\frac{\alpha w}{mw+\theta w} \biggr] \\ &=u \biggl[r-\frac{r}{K}u-\frac{\alpha}{m+\theta} \biggr] \\ &\leq0. \end{aligned}

Therefore, $$\lim_{t \rightarrow\infty} u=0$$ on Ω̅. Consequently, v and w go to zero as $$t \rightarrow\infty$$.

(iii) By assumption, w goes to zero as $$t\rightarrow \infty$$. Also, u and v go to zero as $$t\rightarrow \infty$$ since $$(u,v,w)$$ is contained in Σ.

(iv) Adding the first and second equations in (1.1) and using the facts that $$u+v\leq\theta w$$ and $$\beta\geq\alpha$$ imply

\begin{aligned} (u+v)_{t} -d\Delta(u+v) &=u f_{1} (u,v,w) + v f_{2} (u,v,w) \\ &\leq u \biggl(r -\frac{r}{K}u \biggr) -\frac{\alpha}{m+\theta}u - \frac{\beta}{m+\theta} v -d_{1} v \\ &\leq u \biggl(r + d_{1} -\frac{r}{K}u \biggr) - \biggl( \frac{\alpha }{m+\theta}+d_{1} \biggr) (u+ v) \\ &\leq\frac{K}{r} \biggl(\frac{r+d_{1}}{2} \biggr)^{2} - \biggl(\frac{\alpha}{m+\theta}+d_{1} \biggr) (u+v). \end{aligned}

Thus, $$\limsup_{t\rightarrow\infty} (u +v) \leq\frac{m+\theta}{d_{1} (m+\theta) +\alpha}\frac{K}{r} (\frac{r+d_{1}}{2} )^{2}:=\rho$$, as in Theorem 2.2. Hence, there exists a $$T_{0}$$ such that $$u(t,x)+v(t,x) \leq\rho+\varepsilon$$ on Ω̅ for time $$t\geq T_{0}$$.

Consider the third equation in (1.1):

\begin{aligned} w_{t} -D \Delta w &=w f_{3} (u,v,w) \\ & \leq w \biggl[ \frac{c \beta(u+v)}{mw+u+v} -d_{2} \biggr] \\ & \leq w \biggl[ \frac{(c \beta-d_{2} ) (\rho+\varepsilon)-d_{2} m w}{mw+\rho+\varepsilon} \biggr]. \end{aligned}
(2.5)

Then there exists a $$T_{1} \geq T_{0}$$ such that $$w(t,x) \leq \frac{c\beta-d_{2} }{d_{2} m}(\rho+\varepsilon) +\varepsilon$$ on Ω̅ for time $$t\geq T_{1}$$. Since $$u+v \leq \theta w$$ holds, $$u(t,x) +v(t,x) \leq\theta[\frac{c\beta-d_{2} }{d_{2} m}(\rho+\varepsilon) +\varepsilon]:=\rho(\varepsilon)$$ is satisfied on Ω̅ for time $$t\geq T_{1}$$.

Let $$\tau=\frac{1}{2} [1+\theta\frac{c\beta-d_{2} }{d_{2} m}]$$. Under the assumption that $$\theta< \frac{d_{2} m}{c\alpha-d_{2} }$$, $$\tau<1$$ is satisfied. Since $$\rho(0)=\theta\frac{c\beta-d_{2} }{d_{2} m}\rho<\tau\rho$$, if a sufficiently small $$\varepsilon>0$$ is chosen such that $$\rho(\varepsilon) <\tau\rho$$, $$u(t,x) +v(t,x) \leq\theta[\frac{c\beta-d_{2} }{d_{2} m}(\rho+\varepsilon) +\varepsilon]<\tau\rho$$ on Ω̅ for $$t \geq T_{1}$$.

Now, consider (2.5) under the restriction that $$u(t,x) +v(t,x) \leq\tau\rho$$. Then $$\limsup_{t\rightarrow\infty} w$$ $$\leq\frac{c\beta-d_{2} }{d_{2} m} \tau\rho$$. Thus, there exists a $$T_{2} \geq T_{1}$$ such that $$w(t,x) \leq\frac{c\beta-d_{2} }{d_{2} m} \tau\rho+\varepsilon$$ on Ω̅ for time $$t\geq T_{2}$$. Again, $$u(t,x) +v(t,x) \leq\theta[\frac{c\beta-d_{2} }{d_{2} m}(\tau\rho) +\varepsilon] \leq\tau^{2} \rho$$ on Ω̅ for $$t \geq T_{2}$$ and for a sufficiently small $$\varepsilon>0$$.

Inductively, there exists a sequence $$T_{n}$$ with $$T_{n} \rightarrow \infty$$ such that $$u(t,x) +v(t,x) \leq\tau^{n} \rho$$ on Ω̅ for $$t \geq T_{n}$$. Moreover, since $$\tau<1$$, $$u+v \rightarrow 0$$ uniformly on Ω̅ as $$t\rightarrow\infty$$. Consequently, w goes to zero as $$t \rightarrow\infty$$ as well. □

Asymptotic stability of $$\mathbf{e_{1}}$$

In this subsection, we investigate the stability at $$(K,0,0)$$ under the following conditions:

$$d_{1} \geq bK,\qquad d_{2} \geq c \alpha\quad \mbox{and} \quad r> \frac{\alpha}{m}.$$
(2.6)

The next result implies that only the susceptible prey can survives (Figure 2).

Theorem 2.6

Under assumption (2.6), $$\lim_{t\rightarrow\infty} \mathbf{u} = \mathbf{e_{1}}$$ uniformly on Ω̅.

Proof

From Theorem 2.2, we already know $$\limsup_{t \rightarrow \infty} u \leq K$$. Furthermore, since $$d_{1} \geq bK$$, $$v_{t} -d\Delta v =v f_{2} (u,v,w) \leq0$$ implies $$v \rightarrow0$$ uniformly on Ω̅ as $$t \rightarrow \infty$$. Thus there exists a $$T_{1} >0$$ such that $$v \leq \varepsilon$$ for an arbitrary $$\varepsilon>0$$ and $$t \geq T_{1}$$. Since ε is arbitrary, the assumption that $$d_{2} \geq c\alpha$$ and the comparison principle imply $$w \rightarrow0$$ uniformly on Ω̅ as $$t \rightarrow\infty$$. Therefore, there exists $$T_{2} \geq T_{1}$$ such that $$w \leq\varepsilon$$ for $$t \geq T_{2}$$.

Note that $$\liminf_{t\rightarrow\infty} u\geq (r-b\varepsilon-\frac{\alpha}{m})\frac{K}{r}:=\Theta$$ can also be obtained using the methods from Theorem 2.3. Since $$uf_{1} (u,v,w) \geq u[r-\frac{r}{K}u-\frac{\alpha \varepsilon}{m\varepsilon+u}-b\varepsilon] \geq u[r-\frac{r}{K}u-\frac{\alpha \varepsilon}{m\varepsilon+\Theta-\varepsilon}-b\varepsilon]$$ for $$t \geq T_{2}$$, $$\liminf_{t \rightarrow\infty} u \geq (r-b\varepsilon-\frac{\alpha\varepsilon}{(m-1)\varepsilon +\Theta})\frac{K}{r}>\Theta$$ follows from the comparison principle. Therefore, since ε is arbitrary, $$\lim_{t\rightarrow \infty} \mathbf{u} =e_{1}$$ uniformly on Ω̅. □

Asymptotic stability of $$\mathbf{e_{2}}$$

We investigate the stability at $$(K(1-\frac{c\alpha-d_{2}}{cmr}), 0, \frac{c\alpha -d_{2}}{d_{2} m}K(1-\frac{c\alpha-d_{2}}{cmr}) )$$ under the following condition:

$$rmd_{2} \leq(rm-\alpha) (c\alpha-d_{2} ),\qquad bK < d_{1} \quad\mbox{and}\quad 0< c\alpha-d_{2} < cmr.$$
(2.7)

For simplicity, let $$u_{2}^{*} =K(1-\frac{c\alpha-d_{2}}{cmr})$$ and $$w_{2}^{*} = \frac{c\alpha -d_{2}}{d_{2} m}u_{2}^{*}$$.

The following theorem indicates that one can control the infected prey, namely, only the infected prey can be removed out under some conditions (Figure 3).

Theorem 2.7

Under assumption (2.7), $$\lim_{t\rightarrow\infty} \mathbf{u} =\mathbf{e_{2}}$$ uniformly on Ω̅.

Proof

We prove this theorem by induction. First, consider the following parabolic problem:

$$\textstyle\begin{cases} {\overline{u}_{1}}_{t} -d\Delta{\overline{u}_{1}} = {\overline{u}_{1}}(r-\frac{r}{K}{\overline{u}_{1}}) \quad \mbox{in (0,\infty)\times\Omega,} \\ \frac{\partial{\overline{u}_{1}}}{\partial\eta}=0 \quad \mbox{on (0,\infty)\times\partial \Omega,} \\ {\overline{u}_{1}}(0,x)=u_{0} (x)\quad \mbox{in \Omega.} \end{cases}$$

Then there exists a $$T_{1}^{1} >0$$ such that $$u \leq\overline{ u}_{1}^{*} (\equiv K)+\varepsilon$$ on Ω̅ for $$t \geq T_{1}^{1}$$ and a sufficiently small ϵ such that $$\frac{d_{1} }{b}-K>\varepsilon>0$$.

Next, consider the following problem under the condition that $$bK < d_{1}$$:

$$\textstyle\begin{cases} {\overline{v}}_{t} -d\Delta{\overline{v}} = {\overline{v}}(b(\overline {u}_{1}^{*} +\varepsilon) -d_{1} ) \quad \mbox{in (T_{1}^{1},\infty)\times\Omega,} \\ \frac{\partial{\overline{v}}}{\partial\eta}=0 \quad \mbox{on (T_{1}^{1},\infty)\times\partial \Omega,} \\ {\overline{v}}(0,x)=v (T_{1}^{1},x)\quad \mbox{in \Omega.} \end{cases}$$

Then there exists a $$T_{2}^{1} \geq T_{1}^{1}$$ such that $$v \leq \varepsilon$$ on Ω̅ for $$t \geq T_{2}^{1}$$.

Consider the following problem:

$$\textstyle\begin{cases} {\underline{u}_{1}}_{t} -d\Delta{\underline{u}_{1}} = {\underline{u}_{1}}(r-\frac{\alpha}{m}-b\varepsilon-\frac {r}{K}{\underline{u}_{1}}) \quad \mbox{in (T_{2}^{1},\infty)\times\Omega,} \\ \frac{\partial{\underline{u}_{1}}}{\partial\eta}=0 \quad \mbox{on (T_{2}^{1},\infty)\times\partial \Omega,} \\ {\underline{u}_{1}}(0,x)=u_{0} (x)\quad \mbox{in \Omega.} \end{cases}$$

Then there exists a $$T_{3}^{1} \geq T_{2}^{1}$$ such that $$u \geq \underline{u}_{1}^{*} (\equiv\frac{K}{r}(r-\frac{\alpha}{m})) -(1+\frac{K}{r})\varepsilon$$ on Ω̅ for $$t \geq T_{3}^{1}$$. For simplicity, and since the choice of ε does not affect our proof, redefine $$(1+\frac{K}{r})\varepsilon$$ by $$\varepsilon>0$$.

Consider the following problem:

$$\textstyle\begin{cases} {\underline {w}_{1}}_{t} -D\Delta{\underline{w}_{1}} = \underline {w}_{1}(\frac{c\alpha(\underline{u}_{1}^{*} -\varepsilon)}{m{\underline{w}_{1}} +\underline {u}_{1}^{*} } -d_{2} ) \quad\mbox{in (T_{3}^{1},\infty)\times\Omega,} \\ \frac{\partial{\underline{w}_{1}}}{\partial\eta}=0 \quad \mbox{on (T_{3}^{1},\infty)\times\partial \Omega,} \\ {\underline{w}_{1}}(0,x)=w (T_{3}^{1},x)\quad \mbox{in \Omega.} \end{cases}$$

Then there exists a $$T_{4}^{1} \geq T_{3}^{1}$$ such that $$w \geq \underline{w}_{1}^{*} (\equiv\frac{c\alpha-d_{2} }{d_{2} m}\underline{ u}_{1}^{*}) -\varepsilon$$ on Ω̅ for $$t \geq T_{3}^{1}$$.

Consider the following problem:

$$\textstyle\begin{cases} {\overline{w}_{1}}_{t} -D\Delta{\overline{w}_{1}} = {\overline{w}_{1}}(\frac{c\alpha(\overline{ u}_{1}^{*} +\varepsilon)}{m{\overline{w}_{1}} +\overline{ u}_{1}^{*} +\varepsilon} -d_{2} +\frac{c\beta\varepsilon}{m(\underline {w}_{1}^{*} -\varepsilon)+\varepsilon} ) \quad \mbox{in (T_{4}^{1},\infty)\times\Omega,} \\ \frac{\partial{\overline{w}_{1}}}{\partial\eta}=0\quad \mbox{on (T_{4}^{1},\infty)\times\partial \Omega,} \\ {\overline{w}_{1}}(0,x)=w (T_{4}^{1},x)\quad \mbox{in \Omega.} \end{cases}$$

Then there exists a $$T_{5}^{1} \geq T_{4}^{1}$$ such that $$w \leq \overline{ w}_{1}^{*} (\equiv\frac{c\alpha-d_{2}}{d_{2} m} \overline{u}_{1}^{*} )+\varepsilon$$ on Ω̅ for $$t \geq T_{5}^{1}$$.

Consequently, for $$t \geq T^{1} \equiv T_{5}^{1}$$ and $$x \in\overline{\Omega}$$, the relation

\begin{aligned} &\underline{ u}_{1}^{*} -\varepsilon\leq u \leq\overline{u}_{1}^{*} +\varepsilon, \\ &0 \leq v \leq\varepsilon, \\ &\underline {w}_{1}^{*} -\varepsilon\leq w \leq\overline{w}_{1}^{*} +\varepsilon \end{aligned}

are satisfied.

For induction, consider the following problems for $$T^{n-1} \leq T_{1}^{n-1} \leq T_{2}^{n-1} \leq T_{3}^{n-1} \leq T_{4}^{n-1}$$ and $$n \geq2$$:

\begin{aligned} &\textstyle\begin{cases} {\overline{u}_{n}}_{t} -d\Delta{\overline{u}_{n}} = {\overline{u}_{n}}(r- \frac{\alpha(\underline {w}_{n-1}^{*} -\varepsilon)}{m(\underline {w}_{n-1}^{*} -\varepsilon) + \overline{ u}_{n-1}^{*} +\varepsilon} - \frac{r}{K}{\overline{u}_{n}}) \quad\mbox{in (T^{n-1},\infty)\times \Omega,} \\ \frac{\partial{\overline{u}_{n}}}{\partial\eta}=0 \quad \mbox{on (T^{n-1},\infty)\times\partial \Omega,} \\ {\overline{u}_{n}}(0,x)=u(T^{n-1},x)\quad \mbox{in \Omega,} \end{cases}\displaystyle \\ &\textstyle\begin{cases} {\overline{v}_{n}}_{t} -d\Delta{\overline{v}_{n}} = {\overline{v}_{n}}(b(\overline {u}_{n}^{*} +\varepsilon) -d_{1} ) \quad \mbox{in (T_{1}^{n-1},\infty)\times\Omega,} \\ \frac{\partial{\overline{v}_{n} }}{\partial\eta}=0 \quad \mbox{on (T_{1}^{n-1},\infty)\times\partial \Omega,} \\ {\overline{v}_{n} }(0,x)=v (T_{1}^{n-1},x)\quad \mbox{in \Omega,} \end{cases}\displaystyle \\ &\textstyle\begin{cases} {\overline{w}_{n}}_{t} -D\Delta{\overline{w}_{n}} = {\overline{w}_{n}}(\frac{c\alpha(\overline {u}_{n}^{*} +\varepsilon)}{m{\overline{w}_{n}} +\overline {u}_{n}^{*} +\varepsilon} -d_{2} +\frac{c\beta\varepsilon}{m(\underline {w}_{1}^{*} -\varepsilon)+\varepsilon} ) \quad\mbox{in (T_{2}^{n-1},\infty)\times\Omega,} \\ \frac{\partial{\overline{w}_{n}}}{\partial\eta}=0 \quad \mbox{on (T_{2}^{n-1},\infty)\times\partial \Omega,} \\ {\overline{w}_{n}}(0,x)=w (T_{2}^{n-1},x)\quad\mbox{in \Omega,} \end{cases}\displaystyle \\ &\textstyle\begin{cases} {\underline{u}_{n}}_{t} -d\Delta{\underline{u}_{n}} = {\underline{u}_{n}}(r-\frac{\alpha(\overline{ w}_{n}^{*} +\varepsilon)}{m(\overline{ w}_{n}^{*} +\varepsilon) + \underline {u}_{n-1}^{*} }-b\varepsilon-\frac{r}{K}{\underline{u}_{n}}) \quad \mbox{in (T_{3}^{n-1},\infty)\times\Omega,} \\ \frac{\partial{\underline{u}_{n}}}{\partial\eta}=0 \quad\mbox{on (T_{3}^{n-1},\infty)\times\partial \Omega,} \\ {\underline{u}_{n}}(0,x)=u(T_{3}^{n-1},x)\quad \mbox{in \Omega,} \end{cases}\displaystyle \\ &\textstyle\begin{cases} {\underline{w}_{n}}_{t} -D\Delta{\underline{w}_{n}} = {\underline{w}_{n}}(\frac{c\alpha(\underline {u}_{n}^{*} -\varepsilon)}{m{\underline{w}_{n}} + (\underline{ u}_{n}^{*} - \varepsilon )} -d_{2} ) \quad \mbox{in (T_{4}^{n-1},\infty)\times\Omega,} \\ \frac{\partial{\underline{w}_{1}}}{\partial\eta}=0 \quad \mbox{on (T_{4}^{n-1},\infty)\times\partial \Omega,} \\ {\underline{w}_{1}}(0,x)=w (T_{4}^{n-1},x)\quad \mbox{in \Omega.} \end{cases}\displaystyle \end{aligned}

Therefore, for $$t \geq T^{n} \equiv T_{4}^{n-1}$$, $$n \geq2$$ and $$x \in\overline{\Omega}$$,

\begin{aligned} &\underline {u}_{n}^{*} -\varepsilon\leq u \leq\overline {u}_{n}^{*} +\varepsilon, \\ &0 \leq v \leq\varepsilon, \\ &\underline {w}_{n}^{*} -\varepsilon\leq w \leq\overline{ w}_{n}^{*} +\varepsilon, \end{aligned}

where

\begin{aligned} &\overline {u}_{n}^{*} = \frac{K}{r} \biggl(r-\frac{\alpha\underline {w}_{n-1}^{*} }{m\underline {w}_{n-1}^{*} + \overline {u}_{n-1}^{*}} \biggr), \\ &\overline {w}_{n}^{*} = \frac{c\alpha-d_{2} }{d_{2} m} \overline {u}_{n}^{*}, \\ &\underline{ u}_{n}^{*} = \frac{K}{r} \biggl(r-\frac{\alpha\overline {w}_{n}^{*} }{m\overline{ w}_{n}^{*} + \underline {u}_{n-1}^{*}} \biggr), \\ &\underline{ w}_{n}^{*} = \frac{c\alpha-d_{2} }{d_{2} m} \underline {u}_{n}^{*}. \end{aligned}

Note that $$\overline {u}_{n}^{*}$$, $$\overline {w}_{n}^{*}$$, $$\underline{ u}_{n}^{*}$$, and $$\underline {w}_{n}^{*}$$ are all positive constants. Moreover, the following monotonicity holds:

\begin{aligned} &\underline {u}_{1}^{*} \leq\underline {u}_{2}^{*} \leq\cdots \leq\underline {u}_{n}^{*} \leq\cdots\leq{u_{2}}^{*} \leq \cdot \cdot\cdot\overline {u}_{n}^{*} \leq\cdot\cdot\cdot\overline{ u}_{2}^{*} \leq\overline {u}_{1}^{*}, \\ &\underline {w}_{1}^{*} \leq\underline {w}_{2}^{*} \leq\cdots \leq\underline {w}_{n}^{*} \leq\cdots\leq{w_{2}}^{*} \leq \cdot \cdot\cdot\overline{ w}_{n}^{*} \leq\cdot\cdot\cdot\overline{ w}_{2}^{*} \leq\overline {w}_{1}^{*}, \end{aligned}

since $$\frac{\alpha\underline{ w}_{n}^{*} }{m\underline{ w}_{n}^{*} + \overline{ u}_{n}^{*}} \geq \frac{\alpha\underline{ w}_{n-1}^{*} }{m\underline {w}_{n-1}^{*} + \overline {u}_{n-1}^{*}}$$ and $$\frac{\alpha\overline{ w}_{n}^{*} }{m\overline {w}_{n}^{*} + \underline {u}_{n-1}^{*}} \leq\frac{\alpha\overline {w}_{n-1}^{*} }{m\overline {w}_{n-1}^{*} + \underline {u}_{n-2}^{*}}$$ for all n by induction. Also $$\underline{u}_{n}^{*} \leq u_{2}^{*} \leq\overline {u}_{n}^{*}$$ holds for all n, since $$\frac{\alpha \underline {w}_{n-1}^{*} }{m\underline{ w}_{n-1}^{*} + \overline {u}_{n-1}^{*}} \leq \frac{\alpha\frac{c\alpha-d_{2}}{d_{2} m} }{m \frac{c\alpha-d_{2}}{d_{2} m} +1}=\frac{c\alpha-d_{2}}{c m} \leq \frac{\alpha\overline{ w}_{n}^{*} }{m\overline{ w}_{n}^{*} + \underline {u}_{n-1}^{*}}$$ for $$\frac{\overline {u}_{n}^{*}}{\underline {u}_{n-1}^{*}} \geq1$$ and by the definitions of $$\overline{ w}_{n}^{*}$$ and $$\underline{ w}_{n}^{*}$$. It follows that $$\underline {w}_{n}^{*} \leq w_{2}^{*} \leq\overline {w}_{n}^{*}$$ for all n.

Thus, since the constant sequences $$\{\overline {u}_{n}^{*} \}$$ and $$\{\overline {w}_{n}^{*} \}$$ are monotone nonincreasing, and bounded from below, and the sequences $$\{\underline {u}_{n}^{*} \}$$ and $$\{\underline {w}_{n}^{*} \}$$ are monotone nondecreasing, and bounded from above, the limits of these sequences exist. Denote these limits by , , $$\underline{u}$$, and $$\underline{w}$$, respectively. Consequently, $$\underline{u} \leq u_{2}^{*} \leq\overline{u}$$ and $$\underline{w} \leq w_{2}^{*} \leq\overline{w}$$. The following also holds:

$$\textstyle\begin{cases} {\overline{u}} = \frac{K}{r}(r-\frac{\alpha{\underline{w}}}{m{\underline{w}} + {\overline{u}}} ),\\ {\overline{w}} = \frac{c\alpha-d_{2} }{d_{2} m} {\overline{u}},\\ {\underline{u}} = \frac{K}{r}(r-\frac{\alpha{\overline{w}}}{m{\overline{w}} + {\underline{u}}} ),\\ {\underline{w}} = \frac{c\alpha-d_{2} }{d_{2} m} {\underline{u}}. \end{cases}$$
(2.8)

Suppose to the contrary that $$\overline{u} \neq\underline{u}$$. The first and third equations in (2.8) can be rewritten as

\begin{aligned} &r-\frac{r}{K} \overline{u} -\frac{\alpha{ \frac{c\alpha-d_{2} }{d_{2} m} {\underline{u}} }}{m{\frac{c\alpha-d_{2} }{d_{2} m} {\underline{u}}} + {\overline{u}}}=0, \\ &r-\frac{r}{K} \underline{u} -\frac{\alpha{ \frac{c\alpha-d_{2} }{d_{2} m} {\overline{u}} }}{m{\frac{c\alpha-d_{2} }{d_{2} m} {\overline{u}}} + {\underline{u}}}=0, \end{aligned}

respectively. These two equations imply

$$\textstyle\begin{cases} (rm-\alpha)\frac{c\alpha-d_{2}}{d_{2} m} \underline{u} +r \overline{u} -\frac{r}{K} m \frac{c\alpha-d_{2}}{d_{2} m} \overline{u} \underline{u} -\frac{r}{K} \overline{u} \overline{u} &=0,\\ (rm-\alpha) \frac{c\alpha-d_{2}}{d_{2} m} \overline{u} +r \underline{u} -\frac{r}{K} m \frac{c\alpha-d_{2}}{d_{2} m} \overline{u} \underline{u} -\frac{r}{K} \underline{u} \underline{u} &=0. \end{cases}$$
(2.9)

Subtracting the second equation from the first equation in (2.9) yields

$$(\overline{u} -\underline{u}) \biggl(r-\frac{r}{K} (\overline{u} +\underline{u}) -(rm-\alpha)\frac{c\alpha-d_{2}}{d_{2} m} \biggr)=0.$$

By assumption, since $$\overline{u} \neq\underline{u}$$ (i.e., $$\overline{u} >\underline{u}$$), $$A:=r -\frac{r}{K} (\overline{u} +\underline{u}) -(rm-\alpha)\frac{c\alpha-d_{2}}{d_{2} m}$$ must be zero. But $$A< r-(rm-\alpha)\frac{c\alpha-d_{2}}{d_{2} m} \leq0$$ from (2.7). Hence, $$\overline{u} =\underline{u} = u_{2}^{*}$$; likewise, $$\overline{w} =\underline{w} =w_{2}^{*}$$. Consequently, as time t goes to infinity (i.e., $$n \rightarrow\infty$$),

\begin{aligned} &u_{2}^{*} -\varepsilon\leq u \leq u_{2}^{*} +\varepsilon, \\ &0 \leq v \leq\varepsilon, \\ &w_{2}^{*} -\varepsilon\leq w \leq w_{2}^{*} +\varepsilon, \end{aligned}

are satisfied for an arbitrary $$\varepsilon>0$$. Therefore, the desired result is achieved. □

In the following theorem, we modify the condition that $$bK< d_{1}$$ in (2.7) by reversing the inequality, i.e., $$bK>d_{1}$$, since $$bK< d_{1}$$ causes v to converge to zero automatically.

Theorem 2.8

If the following conditions hold:

\begin{aligned} &rmd_{2} \leq(rm-\alpha) (c\alpha-d_{2} ), \\ &d_{1} < bK < d_{1} +\frac{\beta\sigma}{m \sigma+\theta}, \\ &0< c\alpha-d_{2} < cmr, \\ &r>\frac{\alpha}{m} +b \theta, \end{aligned}

where $$\theta:= \frac{1}{d_{1}}\frac{K}{r} (\frac{r+d_{1}}{2} )^{2}$$ and $$\sigma:= \frac{c\alpha-d_{2}}{d_{2} m} (r-\frac{\alpha}{m} -b\theta)\frac{K}{r}$$, then $$\lim_{t\rightarrow\infty} \mathbf{u} = \mathbf{e_{2}}$$ uniformly on Ω̅.

Proof

First, note that there exists a $$T_{1} >0$$ such that $$u \leq K +\varepsilon$$ and $$u+v \leq\theta+\varepsilon$$ for $$t \geq T_{1}$$ and $$x \in\overline{\Omega}$$, as in Theorem 2.2.

Consider the following parabolic problem:

$$\textstyle\begin{cases} U_{t} -d\Delta U = U(r-\frac{\alpha}{m} -b(\theta+\varepsilon)- \frac{r}{K}U) \quad \mbox{in (T_{1},\infty)\times\Omega,} \\ \frac{\partial U}{\partial\eta}=0 \quad \mbox{on (T_{1} ,\infty)\times\partial \Omega,} \\ U(0,x)=u (T_{1},x)\quad \mbox{in \Omega.} \end{cases}$$

Then there exists $$T_{2} \geq T_{1}$$ such that $$u \geq (r-\frac{\alpha}{m} -b\theta)\frac{K}{r} -\varepsilon$$ on Ω̅ for $$t \geq T_{2}$$. It follows that $$w \geq \sigma-\varepsilon$$ for $$t \geq T_{3}$$ and $$x\in\overline{\Omega}$$ where $$T_{3} \geq T_{2}$$. Now, we are ready to prove that $$\lim_{t \rightarrow\infty} v=0$$ uniformly on Ω̅.

Consider the following problem:

$$\textstyle\begin{cases} V_{t} -d\Delta V = V(b(K+\varepsilon) -\frac{\beta(\sigma-\varepsilon)}{m(\sigma -\varepsilon) +\theta+\varepsilon}-d_{1} )\quad \mbox{in (T_{3},\infty)\times\Omega,} \\ \frac{\partial V}{\partial\eta}=0 \quad \mbox{on (T_{3} ,\infty)\times\partial \Omega,} \\ V(0,x)=v (T_{3},x)\quad\mbox{in \Omega.} \end{cases}$$
(2.10)

For a sufficiently small $$\varepsilon>0$$, the right hand side of the first equation in (2.10) is negative because $$bK < d_{1} +\frac{\beta\sigma}{m \sigma+\theta}$$.

Hence, similar to Theorem 2.7, there exists $$T_{4} \geq T_{3}$$ such that $$0 \leq v \leq\varepsilon$$ for $$t \geq T_{4}$$. The remainder of this proof follows using the same argument as Theorem 2.7. □

Asymptotic stability of $$\mathbf{e_{3}}$$

In this subsection, we investigate the stability at $$(\frac{d_{1}}{b}, \frac{1}{b}(r-\frac{r}{K}\frac{d_{1}}{b}),0)$$. Before developing our argument, we define the following notation, which is similar to the notation defined in [34, 35].

Notation 2.9

1. (i)

$$\mu_{i}$$: Eigenvalue of −Δ on Ω under Neumann boundary condition.

2. (ii)

$$E(\mu_{i} )$$: The eigenspace corresponding to $$\mu_{i}$$.

3. (iii)

$$\{ \varphi_{ij}:j=1,\ldots, \dim E(\mu_{i} ) \}$$: An orthonormal basis of $$E(\mu_{i} )$$.

4. (iv)

$$\mathbf{X_{ij}} = \{\mathbf{c} \cdot\varphi_{ij} | \mathbf{c} \in\mathbb{R}^{3}\}$$.

5. (v)

$$\mathbf{X} = \{ \mathbf{u} =(u,v,w) \in[C^{1} (\overline{\Omega})]^{3} | \frac{\partial u}{\partial\eta}=\frac{\partial v}{\partial\eta}=\frac{\partial w}{\partial\eta}=0 \mbox{ on } \partial\Omega \}$$.

The eigenvalues in (i) satisfy $$0=\mu_{1} <\mu_{2} <\mu_{3} <\cdots\rightarrow\infty$$. Also, $$\mathbf{X} =\bigoplus_{i=1}^{\infty} \mathbf{X_{i}}$$, where $$\mathbf{X_{i}} = \bigoplus_{j=1}^{\dim E(\mu_{i} )} \mathbf{X_{ij}}$$. Now, we show the local stability at $$\mathbf{e_{3}}$$.

The susceptible prey and the infected prey may survive together without the predator (Figure 4).

Theorem 2.10

If $$\max\{ \alpha, \beta\} < \frac{d_{2}}{c}$$ and $$d_{1} (r+bK)>b^{2} K^{2} >d_{1}^{2}$$, then equilibrium $$\mathbf{e_{3}}$$ of (1.1) is locally asymptotically stable.

Proof

First, note that the above assumptions guarantee the positiveness of $$\mathbf{e_{3}}$$. For simplicity, let $$u_{3}^{*} =\frac{d_{1}}{b}$$ and $$v_{3}^{*} =\frac{1}{b}(r-\frac{r}{K}u_{3}^{*} )$$. The assumptions that $$\max\{ \alpha, \beta\} < \frac{d_{2}}{c}$$ and $$d_{1} (r+bK)>b^{2} K^{2}$$ also guarantee that the positivity of $$\Theta:=\frac{(d_{2} -c\alpha)u_{3}^{*} +(d_{2} -c\beta)v_{3}^{*}}{u_{3}^{*} +v_{3}^{*}}$$ and $$(\frac{r}{K})^{2} u_{3}^{*} -b^{2} v_{3}^{*}$$.

The linearization of (1.1) is $$\mathbf{u}_{t} =(\mathbf{D} \Delta+\mathbf{F_{u}} (\mathbf{e_{3}}))\mathbf{u}$$ at the constant solution $$\mathbf{e_{3}}$$, where $$\mathbf{u} =(u(t,x), v(t,x), w(t,x))^{T}$$, $$\mathbf{F}=(uf_{1},vf_{2}, wf_{3} )$$,

\begin{aligned} \mathbf{D} ={ \begin{pmatrix} d &0 &0\\ 0 & d &0\\ 0& 0& D \end{pmatrix} } \quad\mbox{and}\quad \mathbf{F_{u} (e_{3} )} ={ \begin{pmatrix} -\frac{r}{K}u_{3}^{*} & -bu_{3}^{*} &-\frac{\alpha u_{3}^{*}}{u_{3}^{*} +v_{3}^{*}}\\ bv_{3}^{*} & 0 &-\frac{\alpha u_{3}^{*}}{u_{3}^{*} +v_{3}^{*}}\\ 0& 0& -\Theta \end{pmatrix} }. \end{aligned}

For $$i \geq1$$, Xi is invariant under the operator $$\mathbf{D} \Delta+\mathbf{F_{u}} (\mathbf{e_{3}})$$. Note that λ is an eigenvalue of this operator on Xi if and only if it is an eigenvalue of the matrix $$-\mu_{i} \mathbf{D} + \mathbf{F_{u}} (\mathbf{e_{3}})$$. The coefficients of the characteristic polynomial $$\det(\lambda\mathbf{I} +\mu_{i} \mathbf{D} - \mathbf{F_{u}} (\mathbf{e_{3}}))$$ are given by $$\lambda^{3} +A_{i} \lambda^{2} +B_{i} \lambda+C_{i}$$, where

\begin{aligned} &A_{i} =(2d +D )\mu_{i} +\frac{r}{K}u_{3}^{*} +\Theta>0, \\ &B_{i} = (d +2D )d \mu_{i}^{2} + \biggl( \biggl( \frac{r}{K}u_{3}^{*} +2\Theta \biggr)d +\frac{r}{K}u_{3}^{*} D \biggr) \mu_{i} + b^{2} u_{3}^{*} v_{3}^{*} +\frac{r}{K}u_{3}^{*} \Theta>0, \\ &C_{i} = d^{2} D \mu_{i}^{3} + \biggl( \frac{r}{K}u_{3}^{*} dD +d^{2} \Theta \biggr) \mu_{i}^{2} + \biggl(2 b^{2} u_{3}^{*} v_{3}^{*} D+ \frac{r}{K} u_{3}^{*} d \Theta \biggr) \mu_{i} +2b^{2} u_{3}^{*} v_{3}^{*} \Theta>0. \end{aligned}

It is easy to verify that $$A_{i}$$, $$B_{i}$$ and $$C_{i}$$ are all positive.

Finally, we obtain $$A_{i} B_{i} -C_{i} = \tau_{i}^{1} \mu_{i}^{3} +\tau_{i}^{2} \mu_{i}^{2} +\tau_{i}^{3} \mu_{i} +\tau_{i}^{4}$$, where

\begin{aligned} &\tau_{i}^{1} = (2d+D) \bigl(d^{2} +2dD \bigr)-d^{2} D=2d(d+D)^{2} >0, \\ &\tau_{i}^{2} =(2d+D) \biggl( \biggl(\frac{r}{K}u_{3}^{*} +2\Theta \biggr)d +\frac{r}{K} u_{3}^{*} D \biggr) +d^{2} \frac{r}{K} u_{3}^{*} +dD \biggl( \frac{r}{K} u_{3}^{*} +2\Theta \biggr)>0, \\ &\tau_{i}^{3} =b^{2} u_{3}^{*} v_{3}^{*} 2d +\frac{r}{K} u_{3}^{*} d\Theta+ 2 \frac{r}{K} u_{3}^{*}\Theta D \\ &\phantom{\tau_{i}^{3} =}{}+ \biggl(\frac{r}{K}u_{3}^{*} +\Theta \biggr) \biggl( \frac{r}{K} +2\Theta \biggr)d + \biggl( \biggl(\frac{r}{K} \biggr)^{2} u_{3}^{*} u_{3}^{*} -b^{2} u_{3}^{*} v_{3}^{*} \biggr)D>0, \\ &\tau_{i}^{4} = \frac{r}{K} b^{2} u_{3}^{*} v_{3}^{*} +\frac{r}{K} u_{3}^{*} \Theta^{2} + \biggl( \biggl(\frac{r}{K} \biggr)^{2} u_{3}^{*} u_{3}^{*} -b^{2} u_{3}^{*} v_{3}^{*} \biggr)\Theta>0. \end{aligned}

Hence, $$A_{i} B_{i} -C_{i} >0$$ for all $$i \geq1$$. From the Routh-Hurwitz criterion for each i, the three roots of $$\lambda^{3} +A_{i} \lambda^{2} +B_{i} \lambda+C_{i} =0$$ have negative real parts since $$A_{i}$$, $$C_{i}$$, and $$A_{i} B_{i} -C_{i} >0$$. The remainder of this proof follows from Theorem 5.1.1 in . □

Asymptotic stability of $$\mathbf{u_{*}}$$

We investigate the asymptotic stability of the positive equilibrium point under (2.2) and the following conditions:

$$\textstyle\begin{cases} mc \geq1, \qquad \alpha=\beta,\\ d_{2}< c\alpha,\\ \max\{\frac{d_{2} b}{c\alpha}, \alpha\frac{c\alpha-d_{2}}{d_{2} m}\frac{b}{d_{1}} \} \leq\frac{r}{K} \leq\frac{bd_{1}}{\alpha} \frac{d_{2} m}{c\alpha-d_{2}}. \end{cases}$$
(2.11)

Here, we can choose numerical values that satisfy condition (2.2) and (2.11), for example, $$r=0.3699$$, $$b=0.430$$, $$d_{1} =0.0291$$, $$K=0.9299$$, $$\alpha=\beta=0.0122$$, $$m=50.3$$, $$c=8.0999$$, and $$d_{2} =0.0069$$.

The final result says that all three species can survives together under specific conditions (Figure 5).

Theorem 2.11

If (2.2) and (2.11) hold, then the equilibrium solution $$\mathbf{u_{*}}$$ of (1.1) is locally asymptotically stable.

Proof

The linearization of (1.1) is $$\mathbf{u}_{t} =(\mathbf{D} \Delta+\mathbf{F_{u}} (\mathbf{u_{*}}))\mathbf{u}$$ at the constant solution $$\mathbf{u_{*}}$$, where $$\mathbf{u} =(u(t,x), v(t,x), w(t,x))^{T}$$, $$\mathbf{F}=(uf_{1},vf_{2}, wf_{3} )$$,

\begin{aligned} \mathbf{D} ={ \begin{pmatrix} d &0 &0\\ 0 & d &0\\ 0& 0& D \end{pmatrix} } \end{aligned}

and

\begin{aligned} \mathbf{F_{u} (u_{*} )} = \begin{pmatrix} u_{*} ( -\frac{r}{K} + \frac{\alpha w_{*}}{(mw_{*} +u_{*} +v_{*} )^{2}} ) & u_{*} (\frac{\alpha w_{*}}{ (mw_{*} +u_{*} +v_{*} )^{2}}-b ) & -\alpha u_{*} (\frac{u_{*} +v_{*}}{ (mw_{*} +u_{*} +v_{*} )^{2}} )\\ v_{*} (\frac{\beta w_{*}}{ (mw_{*} +u_{*} +v_{*} )^{2}}+b ) & (\frac{\beta v_{*} w_{*}}{ (mw_{*} +u_{*} +v_{*} )^{2}} ) & -\beta v_{*} (\frac{u_{*} +v_{*}}{ (mw_{*} +u_{*} +v_{*} )^{2}} )\\ w_{*} (\frac{c\alpha m w_{*} +c (\alpha-\beta) v_{*}}{ (mw_{*} +u_{*} +v_{*} )^{2}} ) & w_{*} (\frac{c\beta m w_{*} +c (\beta-\alpha) u_{*}}{ (mw_{*} +u_{*} +v_{*} )^{2}} ) & -m w_{*} (\frac{c(\alpha u_{*} +\beta v_{*})}{ (mw_{*} +u_{*} +v_{*} )^{2}} ) \end{pmatrix} . \end{aligned}

The following notation is adopted for simplicity:

\begin{aligned} \mathbf{F_{u} (u_{*} )} ={ \begin{pmatrix} L_{11} & L_{12} & L_{13}\\ L_{21} & L_{22} & L_{23}\\ L_{31} & L_{32} & L_{33} \end{pmatrix} }. \end{aligned}
(2.12)

For $$i \geq1$$, Xi is invariant under the operator $$\mathbf{D} \Delta+\mathbf{F_{u}} (\mathbf{u_{*}})$$, and λ is an eigenvalue of this operator on Xi, if and only if it is an eigenvalue of the matrix $$-\mu_{i} \mathbf{D} + \mathbf{F_{u}} (\mathbf{u_{*}})$$. The coefficients of the characteristic polynomial $$\det(\lambda\mathbf{I} +\mu_{i} \mathbf{D} - \mathbf{F_{u}} (\mathbf{u_{*}}))$$ are given by $$\lambda^{3} +A_{i} \lambda^{2} +B_{i} \lambda+C_{i}$$, where

\begin{aligned} &A_{i} =(2d +D )\mu_{i} -L_{11} -L_{22} -L_{33}, \\ &B_{i} = d(d+2D) \mu_{i}^{2} + \bigl(-(L_{11} +L_{22} +2L_{33})d -(L_{11} +L_{22})D \bigr)\mu_{i} +L_{11} L_{22} \\ &\phantom{B_{i} =}{} +L_{22} L_{33} +L_{11} L_{33} -L_{13} L_{31}-L_{32} L_{23} -L_{21} L_{12}, \\ &C_{i} = d^{2} D \mu_{i}^{3} + \bigl(-(L_{11}+L_{22})dD -d^{2} L_{33} \bigr)\mu_{i}^{2} + \bigl( (L_{11}L_{33} +L_{22}L_{33} -L_{13} L_{31} -L_{32} L_{23})d \\ &\phantom{B_{i} =}{} +(L_{11} L_{22}-L_{21} L_{12} )D \bigr)\mu_{i} -L_{11}L_{22}L_{33}-L_{21}L_{13}L_{32} -L_{31} L_{12}L_{23}+L_{13}L_{31}L_{22} \\ &\phantom{B_{i} =}{} +L_{32}L_{23}L_{11}+L_{21}L_{12}L_{33} -L_{22}L_{33}L_{33} +L_{13}L_{21}L_{32} +L_{31}L_{12}L_{23}. \end{aligned}

We now verify that the coefficients $$A_{i}$$, $$B_{i}$$ and $$C_{i}$$ are positive under assumption (2.11). In particular,

\begin{aligned} &{-}L_{11} -L_{22} -L_{33}=\frac{r}{K}u_{*} + \frac{(mc-1)\alpha w_{*} ( u_{*} + v_{*} )}{ (mw_{*} +u_{*} +v_{*} )^{2}}>0, \\ &{-}L_{11} -L_{22} = \frac{r}{K}u_{*} - \frac{\alpha u_{*} w_{*}}{(mw_{*} +u_{*} +v_{*} )^{2}} -\frac{\beta v_{*} w_{*}}{ (mw_{*} +u_{*} +v_{*} )^{2}} \\ &\phantom{{-}L_{11} -L_{22} }=\frac{r}{K}u_{*} - \frac{\alpha(u_{*} +v_{*} )}{mw_{*} +u_{*} +v_{*} } \frac{w_{*} }{mw_{*} +u_{*} +v_{*} } \\ &\phantom{{-}L_{11} -L_{22} }=\frac{r}{K}u_{*} - \frac{d_{2}}{c} \frac{bu_{*} -d_{1}}{\alpha} \\ &\phantom{{-}L_{11} -L_{22} }= \biggl(\frac{r}{K} - \frac{d_{2} b}{c\alpha} \biggr)u_{*} + \frac{d_{2} d_{1}}{c\alpha}>0, \end{aligned}

since $$\frac{\alpha(u_{*} +v_{*} )}{mw_{*} +u_{*} +v_{*} }=\frac{d_{2}}{c}$$ and $$\frac{w_{*} }{mw_{*} +u_{*} +v_{*} }=\frac{bu_{*} -d_{1}}{\alpha}$$. Also, $$-L_{11} -L_{22} -2L_{33}>0$$ since $$L_{33}<0$$.

Since $$w_{*} =\frac{c\alpha-d_{2}}{d_{2} m} (u_{*} +v_{*})$$, $$b(u_{*} +v_{*} ) =r-\frac{r}{K}u_{*} +d_{1}$$ and $$r-\frac{r}{K}u_{*} >0$$,

\begin{aligned} & L_{11} L_{22} +L_{22} L_{33} +L_{11} L_{33} -L_{13} L_{31}-L_{32} L_{23} -L_{21} L_{12} \\ &\quad = b^{2} u_{*} v_{*} +\frac{r}{K}\alpha\frac{u_{*} w_{*}}{ (mw_{*} +u_{*} +v_{*} )^{2}} \bigl(mcw_{*} +(mc-1)v_{*} \bigr)>0, \\ &L_{11}L_{33} +L_{22}L_{33} -L_{13} L_{31} -L_{32} L_{23}= \frac{r}{K} mc u_{*} \alpha\frac{(u_{*} +v_{*} ) w_{*}}{ (mw_{*} +u_{*} +v_{*} )^{2}}>0, \\ &L_{11} L_{22}-L_{21} L_{12}\\ &\quad=u_{*} v_{*} \biggl(-\alpha\frac{r}{K} \frac{ w_{*}}{ (mw_{*} +u_{*} +v_{*} )^{2}} +b^{2} \biggr) \\ &\quad=u_{*} v_{*} \biggl(-\alpha\frac{r}{K} \frac{c\alpha-d_{2}}{d_{2} m} \frac{u_{*} +v_{*}}{ (mw_{*} +u_{*} +v_{*} )^{2}} +b^{2} \biggr) \\ &\quad\geq u_{*} v_{*} \biggl(-\alpha\frac{r}{K} \frac{c\alpha-d_{2}}{d_{2} m} \frac{1}{ u_{*} +v_{*}} +b^{2} \biggr) \\ &\quad= u_{*} v_{*} \biggl(-\alpha\frac{r}{K} \frac{c\alpha-d_{2}}{d_{2} m} \frac{b}{r-\frac{r}{K}u_{*} +d_{1}} +b^{2} \biggr) \\ &\quad >u_{*} v_{*} b \biggl(-\alpha\frac{r}{K} \frac{c\alpha-d_{2}}{d_{2} m} \frac{1}{d_{1}} +b \biggr)>0 \end{aligned}

and

\begin{aligned} &{-}L_{11}L_{22}L_{33}-L_{21}L_{13}L_{32} -L_{31} L_{12}L_{23}+L_{13}L_{31}L_{22}+L_{32}L_{23}L_{11}+L_{21}L_{12}L_{33} \\ &\quad =c\alpha b^{2} m u_{*} v_{*} w_{*} \frac{u_{*} +v_{*}}{ (mw_{*} +u_{*} +v_{*} )^{2}}>0. \end{aligned}

It follows that $$A_{i} B_{i} -C_{i} = \tau_{i}^{1} \mu_{i}^{3} +\tau_{i}^{2} \mu_{i}^{2} +\tau_{i}^{3} \mu_{i} +\tau_{i}^{4}$$, where

\begin{aligned} \tau_{i}^{1}={} &2d \bigl((2d+D)D +d^{2} \bigr), \\ \tau_{i}^{2} ={}&d^{2} (-3L_{11} -3 L_{22} -4 L_{33})+D^{2} (-L_{11}-L_{22})+4dD(-L_{11}-L_{22}-L_{33})>0, \\ \tau_{i}^{3} ={}& d \bigl[(L_{11} L_{22} +L_{22} L_{33} +L_{11} L_{33} -L_{13} L_{31}-L_{32} L_{23} -L_{21} L_{12}) +(L_{11}L_{22}-L_{21}L_{12}) \bigr] \\ &{} +d(-L_{11}-L_{22}-L_{33}) (-L_{11}-L_{22}-2L_{33})+D(-L_{11}-L_{22}-L_{33}) (-L_{11}-L_{22}) \\ &{} +D \bigl[(-L_{11}-L_{22}) (-L_{33})+(-L_{32}L_{23})+(-L_{13}L_{31}) \bigr]>0, \\ \tau_{i}^{4} ={}&-L_{11}L_{11}L_{22}-L_{11}L_{11}L_{33} +L_{11} L_{13}L_{31}+L_{11}L_{21}L_{12}-L_{11}L_{22}L_{22}-L_{22}L_{22}L_{33} \\ &{} -L_{11}L_{22}L_{33}+L_{22}L_{32}L_{23} +L_{21} L_{22}L_{12}-L_{11}L_{22}L_{33}-L_{11}L_{33}L_{33}+L_{13}L_{31}L_{33} \\ &{} +L_{32}L_{23}L_{33} -L_{22}L_{33}L_{33} +L_{13}L_{21}L_{32} +L_{31}L_{12}L_{23}. \end{aligned}

The positivity of $$\tau_{i}^{2}$$ and $$\tau_{i}^{3}$$ follows directly from the above calculations. Now, we investigate the sign of $$\tau_{i}^{4}$$ for $$L_{23}L_{32}=L_{22}L_{33}$$. Note that

\begin{aligned} -L_{11} ={}&u_{*} \biggl( \frac{r}{K} - \frac{\alpha w_{*}}{(mw_{*} +u_{*} +v_{*} )^{2}} \biggr) \\ ={}&u_{*} \biggl( \frac{r}{K} - \alpha\frac{c \alpha-d_{2}}{d_{2} m} \frac{u_{*} +v_{*}}{(mw_{*} +u_{*} +v_{*} )^{2}} \biggr) \\ >{}&u_{*} \biggl( \frac{r}{K} - \alpha\frac{c \alpha-d_{2}}{d_{2} m} \frac{1}{u_{*} +v_{*}} \biggr) \\ ={}&u_{*} \biggl( \frac{r}{K} - \alpha\frac{c \alpha-d_{2}}{d_{2} m} \frac{b}{r-\frac{r}{K}u_{*} +d_{1}} \biggr) \\ >{}&u_{*} \biggl( \frac{r}{K} - \alpha\frac{c \alpha-d_{2}}{d_{2} m} \frac{b}{d_{1}} \biggr) \geq0, \end{aligned}

since $$w_{*} =\frac{c\alpha-d_{2}}{d_{2} m} (u_{*} +v_{*})$$, $$b(u_{*} +v_{*} ) =r-\frac{r}{K}u_{*} +d_{1}$$, and $$r-\frac{r}{K}u_{*} >0$$. Finally, since $$L_{31} =L_{32}$$, we obtain the positivity of

\begin{aligned} \tau_{i}^{4} ={}&(-L_{11}-L_{22}) (L_{11}L_{22}-L_{12}L_{21})+(-L_{11}) (-L_{33}) (-L_{11}-L_{22}) \\ &{}+(-L_{11})L_{33}(-L_{22}-L_{33}) + \Upsilon>0, \end{aligned}

where

\begin{aligned} \Upsilon={}& L_{13}L_{31}L_{11} +L_{13}L_{31}L_{33} +L_{13}L_{21}L_{32}+L_{31}L_{12}L_{23} \\ ={}&L_{31} \biggl( (mc-1)\alpha^{2} (u_{*} +v_{*} ) \frac{u_{*} v_{*} w_{*}}{(mw_{*} +u_{*} +v_{*} )^{4}} +(mc-1)\alpha^{2} (u_{*} +v_{*} ) \frac{u_{*}^{2} w_{*}}{(mw_{*} +u_{*} +v_{*} )^{4}} \\ &{} + \frac{\alpha(u_{*} +v_{*} )}{(mw_{*} +u_{*} +v_{*} )^{2}}(-L_{11}) \biggr)>0. \end{aligned}

Hence, $$A_{i} B_{i} -C_{i} >0$$ for all $$i \geq1$$. From the Routh-Hurwitz criterion for each i, the three roots of $$\lambda^{3} +A_{i} \lambda^{2} +B_{i} \lambda+C_{i} =0$$ have negative real parts since $$A_{i}$$, $$C_{i}$$, and $$A_{i} B_{i} -C_{i} >0$$. The remainder of this proof follows from Theorem 5.1.1 in . □

Conclusion

A diffusive predator-prey model with a ratio-dependent functional response and infected prey population was investigated under homogeneous Neumann boundary conditions. We showed that depending on initial data, all species can become extinct if the predation rate is small and the searching efficiency constant of the predation rate of the predator for the susceptible prey is large; in other words, the predator overeats the susceptible prey. On the other hand, we showed that the infected prey becomes extinct if the death rate of the infected prey is sufficiently large without respect to the initial data. Furthermore, the same conclusion holds even if the death rate of the infected prey is relatively small. In , the authors proposed a model by considering that the encounter infection rate is meaningful only in the case that it follows the law of ratio-dependence and not the law of simple mass action. They showed that the model exhibits parasite-induced host extinction. Such an extinction is similar to that induced by a ratio-dependent predator-prey functional response. The stability of the disease-free equilibrium $$\mathbf{e_{2}}$$ implies that under certain conditions, total extinction is not possible and the introduction of infected prey into the system may act as a biological control to save the ecosystem from extinction.

In the case that the searching efficiency constants of the predation rate for the susceptible and infected prey are the same, if the maximum per capita capturing rate of the predator for the susceptible prey is small, i.e., the predation rate is sufficiently large, then the positive equilibrium point is locally asymptotically stable.

As regards application, the model with ratio-dependent functional response in this study can be used and improved to describe the interaction among the diseased-species in ecosystems, the susceptible species, and additional species with a certain biological property.

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Acknowledgements

This work was supported by a Korea University Grant.

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Correspondence to Inkyung Ahn.

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The authors declare that they have no competing interests.

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All authors equally made contributions and approved the final version of this manuscript.

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