One of dominant themes in both ecology and mathematical ecology is the dynamic relationship between predators and their prey due to its universal existence and importance in population dynamics. The investigations on predator-prey models were developed during these thirty years, and more realistic models were derived in view of laboratory experiments and observations (see [1-21] and the references therein). In [22, 23], Nindjin et al. considered a predator-prey model incorporating a modified version of Leslie-Gower functional response as well as the Holling-type Ⅱ functional response

$\left\{\begin{array}{lll} \frac{d x}{d t}=x(a_{1}-bx-\frac{c_{1}y}{x+k_{1}}), \\ \frac{d y}{d t}=y(a_{2}-\frac{c_{2}y}{x+k_{2}}), \\ \end{array}\right.$

(1.1)

where (1.1) is considered associated with initial conditions $x(0)>0, y(0)>0$.

Model (1.1) describes a prey population x which serves as food for a predator with population y. The parameters $a_{1}, a_{2}, b, c_{1}, c_{2}, k_{1}, k_{2}$ are assumed to be only positive values: $a_{1}$ and $a_{2}$are the growth rate of prey x and predator y respectively, b measures the strength of competition among individuals of species x, $c_{1}$ is the maximum value of the per capita reduction rate of x due to y, $k_{1}$ and $k_{2}$ measure the extent to which environment provides protection to prey x and to predator y respectively, and $c_{2}$ has a similar meaning as $c_{1}$.

Recently, Tian and Weng [24] studied the following diffusive predator-prey system with modified Leslie-Gower functional response

$\left\{\begin{array}{lll} \frac{\partial u}{\partial t}=D\triangle u + u (1-u-\frac{\beta_{1}w^{2}}{u^{2}+k_{1}}),&x\in \Omega, t>0, \\ \frac{\partial w}{\partial t}=\triangle w +\alpha w(1-\frac{\beta_{2}w}{u+k_{2}}),&x\in \Omega, t>0, \\ \frac{\partial u}{\partial \bf n }=\frac{\partial w}{\partial \bf n}=0,&x\in \partial\Omega, t>0, \\ u(x, 0)=u_{0}(x)\geq 0, w(x, 0)=w_{0}(x)\geq 0,&x\in \Omega, \\ \end{array}\right.$

(1.2)

where $\bf n$ is the outward unit normal vector of the boundary $\partial\Omega$, $u_{0}, w_{0}$ are continuous functions of x. The intactness of the second equation of non-spatial system (1.2) contains a modified Leslie-Gower term $\frac{\alpha\beta_{2}w}{u}$. If the growth of the predator population is of logistic form, then

$\frac{dw}{dt}=w(\alpha-\frac{w}{C}), $

here C measures the carry capacity set by the environmental resources. Let $C=\frac{u}{\alpha\beta_{2}}$, where $\frac{1}{\alpha\beta_{2}}$ is the conversion factor of prey into predators, then the above equation becomes

$\frac{dw}{dt}=w(\alpha-\frac{\alpha\beta_{2}w}{u}), $

and the term $\frac{\alpha\beta_{2}w}{u}$ measures the loss in the predator population due to the rarity (per capita $w/u$) of its favorite food. If this favorite food is lacking severely, the predator w will switch to other population, but its growth will be limited. By adding a positive constant to the denominator, the equation becomes

$\frac{dw}{dt}=w(\alpha-\frac{\alpha\beta_{2}w}{u+k_{2}}).$

For more biological background of system (1.2), one could refer to [24] and the references cited therein.

Obviously, $E_{1}=(0, 0), E_{2}=(1, 0), E_{3}=(0, \frac{k_{2}}{\beta_{2}})$ are the three trivial equilibrium of (1.2). Moreover, we can easily obtain that system (1.2) has a unique interior equilibrium $E_{4}=(u^{*}, w^{*})$ (i.e., $u^{*}>0, w^{*}>0$) if

$(1-\frac{\beta_{1}}{\beta_{2}^{2}})^{2}<3(k_{1}+\frac{2\beta_{1}k_{2}}{\beta_{2}^{2}})$

(1.3)

and

$k_{1}-\frac{\beta_{1}k_{2}^{2}}{\beta_{2}^{2}}>0$

(1.4)

hold.

Remark 1.1  We should point out that there is a print error in [24]. $(1-\frac{\beta_{1}}{\beta_{2}^{2}})^{2}>3(k_{1}+\frac{2\beta_{1}k_{2}}{\beta_{2}^{2}})$ (see (1.6) in [24]) should be $(1-\frac{\beta_{1}}{\beta_{2}^{2}})^{2} < 3(k_{1}+\frac{2\beta_{1}k_{2}}{\beta_{2}^{2}}).$

Remark 1.2  If we only prove the existence of the positive nontrivial equilibrium, an immediate consequence of the proof of the former result is that system (1.2) has a interior equilibrium $E_{4}=(u^{*}, w^{*})$ if (1.4) holds.

As well known, the permanence is an important topics in the study of dynamics for differential equations and population models (see [10, 12]). Tian and Weng [24] obtained sufficient conditions under which system (1.2) is permanent by using comparison techniques. To the best of the authors' knowledge, for the predator-prey system (1.2), whether one could obtain the sufficient and necessary conditions which insure the permanence of system (1.2) or not is still an open problem.

The aim of this paper is, by further developing the analysis technique of Hale and Waltman [25], to obtain sufficient and necessary conditions which ensure the permanence of system (1.2). It is shown that our result supplements and complements one of the main results of Tian and Weng in [24].

For the reaction-diffusion predator-prey system (1.2), the reduced system is an ordinary differential equation of the form

$\left\{\begin{array}{lll} \frac{d u}{d t}= u(1-u-\frac{\beta_{1}w^{2}}{u^{2}+k_{1}}), \\ \frac{d w}{d t}=\alpha w(1-\frac{\beta_{2}w}{u+k_{2}}).\\ \end{array}\right.$

(2.1)

Obviously, the equilibria of (2.1) consist of three trivial critical points $E_{1}=(0, 0), E_{2}=(1, 0), E_{3}=(0, \frac{k_{2}}{\beta_{2}})$ on the boundary of $\Omega =\{(N, P) : N \geq 0, P \geq 0\}$. Moreover, system (2.1) has a nontrivial critical point $E_{4}=(u^{*}, w^{*})$ if (1.4) holds.

The Jacobian matrix of system (2.1) at $E_{1} = (0, 0)$ is

$J_{E_{1}}=\left( \begin{array}{cc}1&0\\ 0&\alpha \end{array}\right ), $

then it is clear that the Jacobian matrix $J_{E_{1}}$ has two eigenvalues $\lambda_{1}=1, \lambda_{2}=\alpha>0$. Hence the equilibria $E_{1} = (0, 0)$ is unstable.

The Jacobian matrix of system (2.1) at $E_{2} = (1, 0)$ is

$J_{E_{2}}=\left( \begin{array}{cc}-1&0\\ 0&\alpha \end{array}\right ), $

then it is clear that the Jacobian matrix $J_{E_{2}}$ has two eigenvalues $\lambda_{1}=-1, \lambda_{2}=\alpha$. Hence the equilibria $E_{2} = (1, 0)$ is a saddle point, the unstable manifolds lie on the w-axis.

The Jacobian matrix of system (2.1) at $E_{3} = (0, \frac{k_{2}}{\beta_{2}})$ is

$J_{E_{3} }=\left( \begin{array}{cc}1-\frac{\beta_{1}}{k_{1}}(\frac{k_{2}}{\beta_{2}})^{2} & 0\\ \frac{\alpha}{\beta_{2}}& -\alpha \end{array}\right ).$

Hence the equilibria $E_{3} = (0, \frac{k_{2}}{\beta_{2}})$ is a saddle point if (1.4) holds, and it is locally asymptotically stable when $1-\frac{\beta_{1}}{k_{1}}(\frac{k_{2}}{\beta_{2}})^{2}\leq 0$.

We first show a well known conclusion on the Logistic equation.

Lemma 3.1  (see [15]) Assume that $u(x, t)$ is defined by

$\left\{\begin{array}{lll} \frac{\partial u}{\partial t}=d_{1}\triangle u + ru(1-\frac{u}{K}),&x\in \Omega, ~ t>0, \\ \frac{\partial u}{\partial \upsilon}=0,&x\in \partial\Omega, ~ t>0, \\ u(x, 0)=u_{0}(x)> 0,&x\in \Omega, \\ \end{array}\right.$

(3.1)

then $\lim\limits_{t\rightarrow \infty}u(x, t)=K$.

From [24], we can obtain that

Theorem 3.1  (see [24]) All the solutions of (1.2) are nonnegative and defined for all t>0. Furthermore, the nonnegative solution $(u(x, t), w(x, t))$ of (1.2) satisfies

$\underset{t\to +\infty }{\mathop{\lim \sup }}\,\begin{matrix} \max \\ x\in \bar{\Omega } \\ \end{matrix}u(x,t)\le 1,\underset{t\to +\infty }{\mathop{\lim \sup }}\,\begin{matrix} \max \\ x\in \bar{\Omega } \\ \end{matrix}w(x,t)\le \frac{1+{{k}_{2}}}{{{\beta }_{2}}}.$

(3.2)

Remark 3.1  An immediate consequence of the proof of the former result is that for a given $\varepsilon>0$,

$ \Lambda=\Big\{(u, w)\in R^{2}:u\in[0, 1+\varepsilon], w\in [0, \frac{1+k_{2}}{\beta_{2}}+\varepsilon]\Big\}$

is an absorbing set for system (1.2). Hence system (1.2) is dissipative.

Now, let us identify an interesting situation where the prey cannot survive.

Theorem 3.2  If $\frac{1}{4} < k_{1} < \beta_{1}(\frac{k_{2}}{\beta_{2}})^{2}$, then $E_{3}=(0, \frac{k_{2}}{\beta_{2}})$ is global asymptotically stable with respect to nonnegative initial functions.

Proof  Let us assume that $k_{1}-\beta_{1}(\frac{k_{2}}{\beta_{2}})^{2} < 0$, we can easily obtain that for small enough $\varepsilon>0$,

$k_{1}-\beta_{1}(\frac{k_{2}}{\beta_{2}}-\varepsilon)^{2} <0.$

(3.3)

From the second equation of the system, there is a $t_{1}>0$ such that

$\begin{array}{lll} \frac{\partial w}{\partial t}\geq\triangle w +\alpha w(1-\frac{\beta_{2}w}{k_{2}}),&x\in \Omega, ~ t>t_{1}, \\ \frac{\partial w}{\partial \bf n}=0,&x\in \partial\Omega, ~ t>t_{1}, \\ w(x, t_{1})> 0,&x\in \Omega.\\ \end{array}$

(3.4)

Consider the corresponding initial value problem

$\begin{array}{lll} z'(t)=\alpha z(1-\frac{\beta_{2}z}{k_{2}}), t>t_{1}, \\ z(t_{1})=\min\limits_{x\in \overline{\Omega}}w(t_{1}, x)>0.\\ \end{array}$

(3.5)

An application of the comparison principle gives

$\liminf_{t\rightarrow +\infty}\min_{x\in \bar{\Omega}}w(x, t)\geq \frac{k_{2}}{\beta_{2}}. $

(3.6)

From (3.6), it follows that for the above $\varepsilon>0$, there exists a T>0, such that $ w(x, t)> \frac{k_{2}}{\beta_{2}}-\varepsilon$ for all $t\geq T, x\in \Omega$.

Therefore from the first equation of system (1.2) and $\frac{1}{4} < k_{1} < \beta_{1}(\frac{k_{2}}{\beta_{2}})^{2}$, we obtain that

$\begin{array}{lll} \frac{\partial u}{\partial t}-D\triangle u &= u (1-u-\frac{\beta_{1}w^{2}}{u^{2}+k_{1}})\leq u (1-u-\frac{\beta_{1}(\frac{k_{2}}{\beta_{2}}-\varepsilon)^{2}}{u^{2}+k_{1}})\\ &=u \Big( \frac{k_{1}-\beta_{1}(\frac{k_{2}}{\beta_{2}}-\varepsilon)^{2}-u(u^{2}-u+k_{1})}{u^{2}+k_{1}}\Big)\\ &=u \Big( \frac{k_{1}-\beta_{1}(\frac{k_{2}}{\beta_{2}}-\varepsilon)^{2} -u[(u-\frac{1}{2})^{2}+k_{1}-\frac{1}{4}])}{u^{2}+k_{1}}\Big) < 0. \end{array}$

(3.7)

Invoking again the comparison principle, we get that $u(x, t)\rightarrow 0$ as $t\rightarrow \infty$. Hence there is $T_{1}>0$ such that $u(x, t) <\varepsilon$ for all $t\geq T_{1}$.

Now, from the second equation of system (1.2), we obtain

$\begin{array}{lll} \frac{\partial w}{\partial t}-\triangle w \leq \alpha w(1-\frac{\beta_{2}w}{k_{2}+\varepsilon}). \end{array}$

(3.8)

By the comparison principle, we get that $w(x, t)\leq z(t)$ for any $t\geq T_{1}$, where $z(t)$ is the solution of the equation $z'(t)=\alpha z(1-\frac{\beta_{2}z}{k_{2}+\varepsilon})$. Finally our assertion follows from the arbitrariness of $\varepsilon$ and the fact that the solutions with positive initial data of the equation $z'(t)=\alpha z(1-\frac{\beta_{2}z}{k_{2}})$ tends exponentially to $z=\frac{k_{2}}{\beta_{2}}$.

From the above discussion and (3.6), we obtain that $E_{3}=(0, \frac{k_{2}}{\beta_{2}})$ is global asymptotically stable.

Remark 3.2  In [24], Tian and Weng obtained the global asymptotical stability of the equilibria $E_{3} = (0, \frac{k_{2}}{\beta_{2}})$ under the assumption that

$\frac{1}{3} < k_{1} < \beta_{1}(\frac{k_{2}}{\beta_{2}})^{2}.$

(3.9)

Inequality (3.9) implies $\frac{1}{4} < k_{1} < \beta_{1}(\frac{k_{2}}{\beta_{2}})^{2}$, but not conversely, for

$\frac{1}{4} < \frac{1}{3} < k_{1} < \beta_{1}(\frac{k_{2}}{\beta_{2}})^{2}.$

Therefore, we have improved the global asymptotical stability condition of [24] for system (1.2). It is shown that these conditions are weaker than those of Tian and Weng [24].

In this section, we will show that any nonnegative solution $(u(x, t), w(x, t))$ of (1.2) lies in a certain bounded region as $t\rightarrow \infty$ for all $x\in \Omega$.

Now, we will summarize some facts contained in Hale and Waltman, see [25], about the permanence for abstract dynamical systems.

Suppose that $\Omega$ is a complete metric space with $\Omega = \Omega_{0} \cup \partial\Omega_{0}$ for an open set $\Omega_{0}$, where $\partial\Omega_{0}$ is the boundary of the set $\Omega_{0}$. We will typically choose $\Omega_{0}$ to be the positive cone in an ordered Banach space. A flow or semiflow on $\Omega$ under which $\Omega_{0}$ and $\partial\Omega_{0}$ are forward invariant is said to be permanent if it is dissipative and if there is a number $\eta > 0$ such that any trajectory starting in $\Omega_{0}$ will be at least a distance $\eta$ from $\partial\Omega_{0}$ for all sufficiently large t. To state a theorem implying permanence we need a few definitions. An invariant set M for the flow or semiflow is said to be isolated if it has a neighborhood U such that M is the maximal invariant subset of U. Let $\omega(\partial\Omega_{0})\subset \partial\Omega_{0}$ denote the union of the sets $\omega(u)$ over $u\in\partial\Omega_{0}$ (this differs from the standard definition of the $\omega$-limit set of a set but it is more convenient for our purposes; see [26] for a discussion). The set $\omega(\Omega_{0})$ is said to be isolated if it has a covering $M = \cup_{k=1}^{n}M_{k}$ of pairwise disjoint, both sets $M_{k}$ which are isolated and invariant with respect to the flow or semiflow both on $\partial\Omega_{0}$ and on $\Omega = \Omega_{0} \cup \partial\Omega_{0}$. The covering M is then called an isolated covering. Suppose $N_{1}$ and $N_{2}$ are isolated invariant sets (not necessarily distinct). The set $N_{1}$ is said to be chained to $N_{2}$ (denoted $N_{1}\rightarrow N_{2}$) if there exists $u\notin N_{1}\cup N_{2}$ with $u \in W^{u}(N_{1}) \cap W^{s}(N_{2})$ (as usual, $W^{u}$ and $W^{s}$ denote the unstable and stable manifolds, respectively). A finite sequence $N_{1}, N_{2}, \cdots, N_{k}$ of isolated invariant sets is a chain if $ N_{1}\rightarrow N_{2} \rightarrow N_{3 }\rightarrow \cdots\rightarrow N_{k}$ (this is possible for $k = 1$ if $N_{1}\rightarrow N_{1}$). The chain is called a cycle if $N_{k} = N_{1}$. The set $\omega(\partial\Omega_{0})$ is said to be acyclic if there exists an isolated covering $ \cup_{k=1}^{n}M_{k}$ such that no subset of $\{M_{k}\}$ is a cycle. We now state a lemma that can be used to establish permanence.

Lemma 4.1  (see [25]) Suppose that $\Omega$ is a complete metric space with $\Omega = \Omega_{0} \cup \partial\Omega_{0}$, where $\Omega_{0}$ is open. Suppose that a semiflow on $\Omega$ leaves both $\Omega_{0}$ and $\partial\Omega_{0}$ forward invariant, maps bounded sets in $\Omega$ to precompact sets for $t > 0$, and is dissipative. If in addition

(1) $\omega(\partial\Omega_{0})$ is isolated and acyclic,

(2) $W^{s}(M_{k})\cap \Omega_{0}=\emptyset$ for all k, where $ \cup_{k=1}^{n}M_{k}$ is the isolated covering used in the definition of acyclicity of $\omega(\partial\Omega_{0})$;

then the semiflow is permanent, i.e., there exist $\eta > 0$ such that any trajectory with initial data in $\Omega_{0}$ will be bounded away from $\partial\Omega_{0}$ by a distance greater than $\eta$ for t sufficiently large.

Now, we show that system (1.2) is permanent. From the viewpoint of biology, this implies that the two species of prey and predator will always coexist at any time and any location of the inhabit domain, no matter what their diffusion coefficients are.

Theorem 4.1  System (1.2) is permanent if and only if (1.4) holds.

Proof  Let us set $F=(F_{1}, F_{2}), U=(u, w)$ and $D={\rm diag}[D, 1]$, where

$F_{1}(u, w)=u (1-u-\frac{\beta_{1}w^{2}}{u^{2}+k_{1}}), F_{2}(u, w)=\alpha w(1-\frac{\beta_{2}w}{u+k_{2}}).$

Henceforth, considering also an initial condition, system (1.2) can be rewritten as

$\left\{\begin{array}{lll} \frac{\partial U}{\partial t}(x, t)=D\triangle U(x, t) + F(U),&x\in \Omega, ~ t>0, \\ \frac{\partial U}{\partial \bf n}(x, t)=0,&x\in \partial\Omega, ~ t>0, \\ U(x, 0)=\varphi(x)=(\varphi_{1}(x), \varphi_{2}(x)),&x\in \Omega.\\ \end{array}\right.$

(4.1)

Denoting by

$\Omega_{0}=\{\varphi\in X_{\Lambda}:\varphi(x)>0, x\in \overline{\Omega}\}, $

where $X_{\Lambda}=\Omega_{0} \cup \partial\Omega_{0}$. Assume that operator $S(t)$ is compact for $t > 0$, which is defined by $[S(t)\varphi](x) = U(x, t; \varphi)$, where $U(x, t; \varphi)$ is the classical solution of system (1.2). Moreover, from Theorem 3.1, it follows that the semiflow $\{S(t)\}_{t\geq 0}$ is pointwise dissipative.

A direct application of the maximum principle shows that $S(t)$ is positively invariant on $\partial\Omega_{0}$, which in turn implies that $S(t)$ is positively invariant on $\Omega_{0}$.

Now let us show that $\omega(\partial\Omega_{0})=\{E_{1}, E_{2}, E_{3}\}$.

First, we consider the equation

$\left\{\begin{array}{lll} \frac{\partial u}{\partial t}=D\triangle u + u (1-u),&x\in \Omega, ~ t>0, \\ \frac{\partial u}{\partial \bf n }=0,&x\in \partial\Omega, ~ t>0, \\ u(x, 0)=\varphi_{1}(x)>0,&x\in \Omega. \end{array}\right.$

(4.2)

There is no loss of generality assuming that $\varphi_{1}(x)> 0$ for all $x \in \overline{\Omega}$. Indeed, if $\varphi_{1}(x)\geq 0$ for all $x \in \overline{\Omega}$ with $\varphi_{1}(x)\not\equiv 0$ and applying the strong maximum principle, we obtain that $u(x, t) > 0$ for all t > 0 and $x \in \overline{\Omega}$. One could change the origin of time to a positive time $t_{1}$ and choose $\varphi_{1}(x) = u(x, t_{1})$.

Let $p(t)$ and $q(t)$ be the solutions of the equation $z' =z(1-z)$ such that $p(0) = m = \min\limits_{x\in \overline{\Omega}}\varphi_{1}(x) > 0 $ and $q(0) = M = \max\limits_{x\in \overline{\Omega}}\varphi_{1}(x) > 0 $, respectively. From the comparison principle, we obtain

$p(t)\leq u(x, t)\leq q(t), t>0, x\in \overline{\Omega}.$

Taking into account that $z(t) = 1$ attracts any positive solution, we conclude that $u(x, t)\rightarrow 1$ as $t \rightarrow\infty$.

If $\varphi_{1}(x)\equiv 0$, we can easily get that $u(x, t)\equiv 0$.

Let us now consider

$\left\{\begin{array}{lll} \frac{\partial w}{\partial t}=\triangle w +\alpha w(1-\frac{\beta_{2}w}{k_{2}}),&x\in \Omega, ~ t>0, \\ \frac{\partial w}{\partial \bf n}=0,&x\in \partial\Omega, ~ t>0, \\ w(x, 0)=\varphi_{2}(x)> 0,&x\in \Omega. \end{array}\right.$

(4.3)

Similarly, by the comparison principle, we obtain that $w(x, t)\rightarrow \frac{k_{2}}{\beta_{2}} $ as $t\rightarrow\infty$.

Henceforth, we conclude that $\omega(\partial\Omega_{0})=\{E_{1}, E_{2}, E_{3}\}$ and it is isolated and acyclic. By choosing $M_{1} = E_{1}$, $M_{2} = E_{2}$ and $M_{3} = E_{3}$, then $M = M_{1}\cup M_{2}\cup M_{3}$ is the covering required by Lemma 4.1.

Taking into account that $\omega(\partial\Omega_{0})$ is positively invariant and the fact that the manifolds of the equilibrium $E_{1}$ lie on $\omega(\partial\Omega_{0})$. Indeed, let us linearize (1.2) about $E_{1}$, obtaining the following system

$\left\{\begin{array}{lll} \frac{\partial u}{\partial t}=D\triangle u +u, \\ \frac{\partial w}{\partial t}=\triangle w +\alpha w, \end{array}\right.$

(4.4)

we obtain that $W^{s}(E_{1})\cap \Omega_{0} = \emptyset$. In order to show that $W^{s}(E_{2})\cap \Omega_{0} = \emptyset$, let us linearize (1.2) about $E_{2}$, obtaining the following system

$\left\{\begin{array}{lll} \frac{\partial u}{\partial t}=D\triangle u - u, \\ \frac{\partial w}{\partial t}=\triangle w +\alpha w, \end{array}\right.$

(4.5)

we get that the unstable manifold to $E_{2}$ point into the region $\Omega_{0}$. Moreover, from the above performed computations we know that the stable manifold corresponding to the equilibrium $E_{2}$ lies on $\partial\Omega_{0}$. This implies that $W^{s}(E_{2})\cap \Omega_{0} = \emptyset$.

Similarly, in order to show that $W^{s}(E_{3})\cap \Omega_{0} = \emptyset$, let us linearize (1.2) about $E_{3}$, obtaining the following system

$\left\{\begin{array}{lll} \frac{\partial u}{\partial t}=D\triangle u +(1-\frac{\beta_{1}}{k_{1}}(\frac{k_{2}}{\beta_{2}})^{2})u, \\ \frac{\partial w}{\partial t}=\triangle w +\frac{\alpha}{\beta_{2}} u-\alpha w, \end{array}\right.$

(4.6)

we get that the unstable manifold to $E_{3}$ point into the region $\Omega_{0}$. Moreover, from the above performed computations we know that the stable manifold corresponding to the equilibrium $E_{3}$ lies on $\partial\Omega_{0}$. This implies that $W^{s}(E_{3})\cap \Omega_{0} = \emptyset$.

Since all hypothesis of Lemma 4.1 are fulfilled, we may conclude that system (1.2) is permanent.

Permanence implies that condition (1.4) is an immediate consequence of Section 2. Indeed, from Section 2 and Theorem 3.2 in Section 3, we can easily obtain that the trivial equilibrium $E_{3}$ of system (1.2) is locally asymptotically stable if $1-\frac{\beta_{1}}{k_{1}}(\frac{k_{2}}{\beta_{2}})^{2}\leq 0$, which is a contradiction. Therefore, we can easily get that system (1.2) is permanent if and only if (1.4) holds.

This concludes the proof of our claim.

Remark 4.1  In [24], Tian and Weng obtained the permanence of system (1.2) under the assumption that

$1-\frac{\beta_{1}}{k_{1}}(\frac{1+k_{2}}{\beta_{2}})^{2}>0.$

(4.7)

Inequality (4.7) implies (1.4), but not conversely, for

$\frac{\beta_{1}}{k_{1}}(\frac{1+k_{2}}{\beta_{2}})^{2}> \frac{\beta_{1}}{k_{1}}(\frac{k_{2}}{\beta_{2}})^{2}.$

(4.8)

Therefore, we have improved the permanence condition of [24] for system (1.2). It is shown that these conditions are weaker than those of Tian and Weng [24].

Remark 4.2  Moreover, in this paper we obtain the sufficient and necessary conditions which insure the permanence of system (1.2) by using the method in [25], but Tian and Weng [24] only obtained sufficient conditions under which system (1.2) is permanent by using comparison techniques. It is shown that our result supplements and complements one of the main results of Tian and Weng in [24].

Remark 4.3  From Theorem 4.1, we can easily obtain that diffusion has no influence on the permanence of system (1.2).