\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2009(2009), No. 157, pp. 1--12.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2009 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2009/157\hfil A non-autonomous three-dimensional population system] {Dynamics of a non-autonomous three-dimensional population system} \author[T. H. Quang, T. V. Ton, N. T. H. Linh\hfil EJDE-2009/157\hfilneg] {Ta Hong Quang, Ta Viet Ton, Nguyen Thi Hoai Linh} % in alphabetical order \address{Ta Hong Quang \newline Faculty of Mathematics, Xuan Hoa Teacher Training University, Vietnam} \email{hongquangta12@yahoo.com} \address{Ta Viet Ton \newline Division of Precision Science \& Technology and Applied Physics, Graduate School of Engineering, Osaka University, Japan} \email{taviet.ton@ap.eng.osaka-u.ac.jp} \address{Nguyen Thi Hoai Linh \newline Vietnam Publishing House for Science and Technology, Vietnam} \email{linha1t@yahoo.com} \thanks{Submitted August 28, 2009. Published December 1, 2009.} \subjclass[2000]{34C27, 34D05} \keywords{Predator-prey model; survival; extinction; persistence; \hfill\break\indent asymptotic stability; Liapunov function} \begin{abstract} In this paper, we study a non-autonomous Lotka-Volterra model with two predators and one prey. The explorations involve the persistence, extinction and global asymptotic stability of a positive solution. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \newtheorem{proposition}[theorem]{Proposition} \allowdisplaybreaks \section{Introduction} The dynamics of Lotka-Volterra models and their permanence, stability, global attractiveness, coexistence, extinction have been studied by several authors. Takeu\-chi and Adachi \cite{TA} showed that some chaotic motions may occur in the model of three species. Krikorian \cite{KR} considered an autonomous system of three species and obtained some results on global boundedness and stability. Korobeinikov and Wake \cite{KW}, Korman \cite{K} investigated a model of two preys, one predator and another one of two predators, one prey with constant coefficients, where direct competition is absent. Ahmad \cite{A3} obtained necessary and sufficient conditions for survival of species which rely on the averages of the growth rates and the interaction of coefficients. Besides, we also refer to \cite{A1,A2,T1,T2}. In this paper, we consider the following Lotka-Volterra model of two predators and one prey \begin{equation}\label{1} \begin{gathered} x_1'(t)=x_1(t)[ a_1(t)-b_{11}(t) x_1(t)-b_{12}(t) x_2(t)-b_{13}(t) x_3(t)],\\ x_2'(t)=x_2(t)[ -a_2(t)+b_{21}(t) x_1(t)-b_{22}(t) x_2(t) -b_{23}(t) x_3(t)],\\ x_3'(t)=x_3(t)[ -a_3(t)+b_{31}(t) x_1(t)-b_{32}(t) x_2(t)-b_{33}(t) x_3(t)], \end{gathered} \end{equation} where $x_i(t)$ represents the population density of species $X_i$ at time $t$ $(i\geq 1)$, $X_1$ is the prey and $ X_2, X_3$ are the predators and they interact with each other. $a_i(t), b_{ij}(t) (1\leq i,j\leq 3) $ are continuous functions on $\mathbb{R}$ that are bounded above and below by some positive constants. At time $t$, $a_1(t)$ is the intrinsic growth rate of $X_1$, and $a_i(t)$ is the death rate of $X_i (i\geq 2)$; $\frac{b_{i1}(t)}{b_{1i}(t)}$ denotes the coefficient in conversion $X_1$ into new individual of the $X_i (i\geq 2)$; $b_{ij}(t)$ measures the amount of competition between $X_i$ and $X_j$ $(i\ne j, i,j\geq 2)$, and $b_{ii}(t) (i\geq 1)$ measures the inhibiting effect of environment on $X_i$. This article is organized as follows. Section 2 provides some definitions and notations. In Section 3, we state some results on invariant set and asymptotic stability for problem \eqref{1}. In Section 4, we assume that the coefficients $b_{ij}(t)$ $(1\leq i,j\leq 3)$ are constants, then we give some inequalities, involving the average of the coefficients, which guarantees persistence of the system. Section 5 is a special case of Section 4 in which the coefficients $a_i(t)$ $(i\geq 1)$ are constants. We also give some inequalities which imply non-persistence; more specifically, extinction of the third species with small positive initial values. \section{Definitions and notation} In this section we introduce some basic definitions and facts which will be used in next sections. Let $\mathbb{R}^3_+=\{(x_1,x_2,x_3) \in \mathbb{R}^3| x_i \geq 0, i\geq1\}$. For a bounded continuous function $g(t)$ on $\mathbb{R}$, we denote $$ g^u= \sup_{t \in \mathbb{R}}\ g(t), \quad g^l=\inf_{t \in \mathbb{R}} g(t). $$ The existence and uniqueness of the global solutions of system \eqref{1} can be found in \cite{XCCC}. From the uniqueness theorem, it is easy to prove the following result. \begin{lemma}\label{lem.1} Both the non-negative and positive cones of $\mathbb{R}^3$ are positively invariant for \eqref{1}. \end{lemma} In the remainder of this paper, for biological reasons, we only consider the solutions $\left(x_1(t), x_2(t), x_3(t)\right)$ with positive initial values; i.e., $x_i(t_0)>0, i\geq 1$. \begin{definition} \label{def2.2} \rm System \eqref{1} is said to be permanent if there exist positive constants $\delta, \Delta$ with $0<\delta<\Delta$ such that $\liminf_{t \to \infty}x_i(t)\geq \delta$, $\limsup_{t \to \infty}x_i(t)\leq \Delta $ for all $ i\ge1$. System \eqref{1} is called persistent if $\limsup_{t \to \infty}x_i(t)>0$, and strongly persistent if $\liminf_{t \to \infty}x_i(t)>0 $ for all $ i\ge1$. \end{definition} \begin{definition} \label{def2.3} \rm A set $A$ is called to be an ultimately bounded region of system \eqref{1} if for any solution $(x_1(t), x_2(t), x_3(t))$ of \eqref{1} with positive initial values, there exists $T_1>0$ such that $(x_1(t), x_2(t), x_3(t)) \in A$ for all $t \geq t_0+T_1$. \end{definition} \begin{definition} \label{def2.4} \rm A bounded non-negative solution $(x_1^*(t), x_2^*(t), x_3^*(t))$ of \eqref{1} is said to be global asymptotic stable solution (or global attractive solution) if any other solution $(x_1(t), x_2(t), x_3(t))$ of \eqref{1} with positive initial values satisfies $$ \lim_{t \to \infty}\sum_{i=1}^3|x_i(t)-x_i^*(t)|=0. $$ \end{definition} \begin{remark} \label{rmk2.5} \rm It is easy to see that if the system \eqref{1} has a global asymptotic stable solution, then so are all solutions of \eqref{1}. \end{remark} \section{The model with general coefficients} Let $\epsilon$ be a positive constant. We put \begin{gather*} M_1^\epsilon=\frac{a_1^u}{b_{11}^l}+\epsilon, \quad M_2^\epsilon= \frac{-a_2^l+b_{21}^u M_1^\epsilon}{b_{22}^l},\\ M_3^\epsilon=\frac{-a_3^l+b_{31}^uM_1^\epsilon}{b_{33}^l},\quad m_1^\epsilon=\frac{a_1^l-b_{12}^uM_2^\epsilon-b_{13}^uM_3^\epsilon}{b_{11}^u},\\ m_2^\epsilon=\frac{-a_2^u+b_{21}^lm_1^\epsilon-b_{23}^uM_3^\epsilon}{b_{22}^u},\quad m_3^\epsilon=\frac{-a_3^u+b_{31}^l m_1^\epsilon-b_{32}^u M_2^\epsilon }{b_{33}^u}, \end{gather*} \begin{equation} \label{3} \begin{gathered} B_1^\epsilon(t)=a_1(t)-2b_{11}(t)m_1^\epsilon-b_{12}(t)m_2^\epsilon-b_{13}(t)m_3^\epsilon+b_{21}(t)M_2^\epsilon+b_{31}(t)M_3^\epsilon,\\ B_2^\epsilon(t)=-a_2(t)+b_{21}(t)M_1^\epsilon-2b_{22}(t)m_2^\epsilon-b_{23}(t)m_3^\epsilon+b_{12}(t)M_1^\epsilon+b_{32}(t)M_3^\epsilon,\\ B_3^\epsilon(t)=-a_3(t)+b_{31}(t)M_1^\epsilon-2b_{33}(t)m_3^\epsilon-b_{32}(t)m_2^\epsilon+b_{13}(t)M_1^\epsilon+b_{23}(t)M_2^\epsilon. \end{gathered} \end{equation} We have the following theorems. \begin{theorem} \label{thm3.1} If $m_i^\epsilon >0$ for all $ i\geq1$, then the set $\Gamma_\epsilon$ defined by \[ %%% \label{2} \Gamma_\epsilon=\{(x_1, x_2, x_3) \in \mathbb{R}^3|\ m_i^\epsilon \leq x \leq M_i^\epsilon, i\geq1\} \] is positively invariant with respect to system \eqref{1}. \end{theorem} \begin{proof} We know that the logistic equation $$ X'(t)=AX(t) [B-X(t)] \quad (A, B \in \mathbb{R}, B \neq 0) $$ has a unique solution \[ X(t) = \frac{BX_0 \exp\{AB(t-t_0)\}}{X_0\exp\{AB(t-t_0)\}+B-X_0}, \] where $X_0=X(t_0)$. We now consider the solution of system \eqref{1} with the initial values $(x^0_1, x^0_2, x^0_3)$ $ \in \Gamma_\epsilon$. By Lemma \ref{lem.1}, we have $x_i(t)>0$ for all $t\geq t_0$ and $i\geq 1$. We have \begin{align*} x_1'(t) &\leq x_1(t)[a_1(t)-b_{11}(t)x_1(t)]\\ &\leq x_1(t)[a_1^u-b_{11}^lx_1(t)]\\ &=b_{11}^lx_1(t)[M_1^0-x_1(t)]. \end{align*} Using the comparison theorem, we obtain that \begin{equation} \label{4} \begin{aligned} x_1(t) &\leq \frac{x^0_1 M_1^0 \exp\{a_1^u(t-t_0)\}}{x^0_1 \big[\exp\{a_1^u(t-t_0)\}-1\big] +M_1^0} \\ &\leq \frac{x^0_1 M_1^\epsilon \exp\{a_1^u(t-t_0)\}}{x^0_1 \big[\exp\{a_1^u(t-t_0)\}-1\big] +M_1^\epsilon}\cdot \end{aligned} \end{equation} Then, it follows from $x^0_1\leq M_1^\epsilon$ that $x_1(t)\leq M_1^\epsilon$ for all $ t \geq t_0$. On the other hand, from $x^0_2\leq M_2^\epsilon$ and $$ x_2'(t) \leq x_2(t) [-a_2^l+b_{21}^u M_1^\epsilon-b_{22}^l x_2(t)] =b_{22}^lx_3(t) [M_2^\epsilon-x_2(t)], $$ it implies that $ x_2(t)\leq M_2^\epsilon $ for all $t \geq t_0$. Similarly, we can prove that $x_3(t)\leq M_3^\epsilon$ for all $ t \geq t_0$. From the above results, we have $$ x_1'(t) \geq x_1(t)[a_1^l-b_{12}^uM_2^\epsilon-b_{13}^uM_3^\epsilon -b_{11}^u x_1(t)]=b_{11}^ux_1(t)[m_1^\epsilon-x_1(t)]. $$ It follows from $x^0_1\geq m_1^\epsilon$ that $$ x_1(t) \geq \frac{m_1^\epsilon x^0_1 \exp\{b_{11}^u m_1^\epsilon(t-t_0)\}}{x^0_1 \big[\exp\{b_{11}^um_1^\epsilon(t-t_0)\}-1\big]+m_1^\epsilon}\geq m_1^\epsilon\quad \text{for all } t \geq t_0. $$ Similarly, it is easy to see that $x_2(t)\geq m_2^\epsilon, x_3(t)\geq m_3^\epsilon$ for all $ t \geq t_0$. The proof is complete. \end{proof} \begin{theorem}\label{thm2.2} If $m_i^\epsilon>0\, (i\geq 1)$, then the set $\Gamma_\epsilon$ is an ultimately bounded region, i.e., system \eqref{1} is permanent. \end{theorem} \begin{proof} From \eqref{4} we have $\limsup_{t\to \infty} x_1(t) \leq M_1^\epsilon$. Thus, there exist $\epsilon>0$ and $t_1\geq t_0$ such that $x_1(t) \leq M_1^\epsilon$ for all $t\geq t_1$. By the same argument in Theorem \ref{thm3.1}, it can be shown that $\limsup_{t\to \infty} x_i(t) \leq M_i^\epsilon $ and $\liminf_{t\to \infty} x_i(t) \geq m_i^\epsilon (i\geq 2)$. Then $\Gamma_\epsilon$ is an ultimately bounded region with a sufficiently small $\epsilon>0$. \end{proof} In the following theorem, we give some conditions which ensure the extinction of the predators \begin{theorem} \label{thm3.3} If $M_i^0<0$ then $\lim_{t\to \infty} x_i(t)=0, i\geq 2$. \end{theorem} \begin{proof} We see that if $M_i^0<0$ then $M_i^\epsilon<0$ with a sufficiently small $\epsilon$. Similarly as in the proof of Theorem \ref{thm3.1}, we get \begin{equation} \label{5} x_i'(t) \leq b_{ii}^lx_i(t) [M_i^\epsilon-x_i(t)]<0, i\geq 2. \end{equation} Therefore, $00$ then $00$ such that $x_i'(t)<-\nu$ for all $t\geq t_0$. It follows $x_i(t)<-\nu (t-t_0)+x_i(t_0)$ and $\lim_{t\to \infty} x_i(t)=-\infty$ which contradicts the inequality $x_i(t)>0$ for all $t\geq t_0$. Hence, $\lim_{t\to \infty} x_i(t)=0$. \end{proof} Now, to consider the global asymptotic stability of a solution, we need the following result, called Barbalat's lemma (see \cite{B}) \begin{lemma}\label{lem.2} Let $h$ be a real number and $f$ be a non-negative function defined on $[h, +\infty)$ such that $f$ is integrable on $[h, +\infty)$ and uniformly continuous on $[h, +\infty)$. Then $\lim_{t \to \infty}f(t)=0$. \end{lemma} \begin{proof} We suppose that $f(t) \not\to 0$ as $t \to \infty$. There exists a sequence $(t_{n}), t_n\geq h$ such that $t_{n} \to \infty$ as $n \to \infty$ and $f(t_{n}) \geq \varepsilon$ for all $n \in \mathbb{N}$. By the uniform continuity of $f$, there exists a $\delta > 0$ such that, for all $n \in \mathbb{N}$ and $t \in [t_{n}, t_{n}+\delta]$, $ |f(t_{n}) - f(t)| \leq \frac{\varepsilon}{2}$. Thus, for all $t \in [t_{n}, t_{n}+\delta]$ and $n \in \mathbb{N}$ we have \[ f(t)=|f(t_{n}) - [f(t_{n})- f(t)]| \geq |f(t_{n})| - |f(t_{n})- f(t)| \geq \varepsilon - \frac{\varepsilon}{2} = \frac{\varepsilon}{2}. \] Therefore, $$ \int_{t_{n}}^{t_{n}+\delta} f(t) dt = \int_{t_{n}}^{t_{n}+\delta} f(t) dt \geq \frac{\varepsilon \delta}{2} > 0 $$ for each $n \in \mathbb{N}$. By the existence of the Riemann integral $\int_{h}^{\infty} f(t) dt$, the left hand side of the above inequality converges to 0 as $n \to \infty$ yielding a contradiction. \end{proof} \begin{theorem} \label{thm3.4} Let $(x_1^*(t), x_2^*(t), x_3^*(t))$ be a solution of system \eqref{1}. If $m_i^\epsilon>0$ and $\limsup_{t\to \infty}B_i^\epsilon(t)<0$ for all $i\geq 1$, then $(x_1^*(t), x_2^*(t), x_3^*(t))$ is globally asymptotically stable. \end{theorem} \begin{proof} From the assumptions, there exists $t_1>t_0$ such that $\sup_{t\geq t_1}B_i^\epsilon(t)<0$, $i\geq 1$. Let $(x_1(t), x_2(t), x_3(t))$ be any solution of positive initial value system \eqref{1}. Since $\Gamma_\epsilon$ is an ultimately bounded region, there exists $T_1>t_1$ such, that for all $t\geq T_1$, \[ (x_1(t), x_2(t), x_3(t)), (x_1^*(t), x_2^*(t), x_3^*(t)) \in \Gamma_\epsilon. \] Now, we consider a Liapunov function defined by $V(t)=\sum_{i=1}^3|x_i(t)-x_i^*(t)|, t\geq T_1$. For brevity, we denote $x_i(t), x_i^*(t), a_i(t)$ and $b_{ij}(t)$ by $x_i$, $x_i^*$, $a_i$ and $b_{ij}$, respectively. A direct calculation of the right derivative $D^+V(t)$ of $V(t)$ along the solution of system \eqref{1} gives \begin{align*} D^{+}V(t)=&\sum_{i=1}^3 \mathop{\rm sgn}(x_i-x_i^{*}) [{x_i}'-{x^*_i}'] \\ =&\mathop{\rm sgn}(x_1-x_1^{*}) [x_1(a_1-\sum_{j=1}^3 b_{1j} x_j)-x_1^* (a_1-\sum_{j=1}^3 b_{1j}x_j^*)] \\ &+\sum_{i=2}^3 \Big[x_i(-a_i+b_{i1}x_1-\sum_{j=2}^3 b_{ij} x_j) \\ &-x_i^*(-a_i+b_{i1}x_1^*-\sum_{j=1}^3 b_{ij}x_j^*)\Big]\mathop{\rm sgn}(x_i-x_i^{*}) \\ =&[a_1-b_{11}(x_1+x_1^*)]|x_1-x_1^*| \\ &-\mathop{\rm sgn}(x_1-x_1^*)\sum_{j=2}^3 b_{1j}(x_1x_j-x_1^*x_j^*) \\ &+\sum_{i=2}^3[-a_i-b_{ii}(x_i+x_i^*)]|x_i-x_i^*| \\ &+\mathop{\rm sgn}(x_2-x_2^*)[b_{21}(x_1x_2-x_1^*x_2^*)-b_{23}(x_2x_3-x_2^*x_3^*)] \\ &+\mathop{\rm sgn}(x_3-x_3^*)[b_{31}(x_1x_3-x_1^*x_3^*)-b_{32}(x_2x_3-x_2^*x_3^*)] \\ %%%%%%%%%%% =&[a_1-b_{11}(x_1+x_1^*)-b_{12}x_2-b_{13}x_3]|x_1-x_1^*| \\ &+[-a_2+b_{21}x_1-b_{22}(x_2+x_2^*)-b_{23}x_3^*]|x_2-x_2^*| \\ &+[-a_3+b_{31}x_1-b_{33}(x_3+x_3^*)-b_{32}x_2^*]|x_3-x_3^*| \\ &-\mathop{\rm sgn}(x_1-x_1^*) \sum_{j=2}^3b_{1j}x_1^*(x_j-x_j^*) \\ &+\mathop{\rm sgn} (x_2-x_2^*)[b_{21}x_2^*(x_1-x_1^*)-b_{23}x_2(x_3-x_3^*)] \\ &+\mathop{\rm sgn} (x_3-x_3^*)[b_{31}x_3^*(x_1-x_1^*)-b_{32}x_3(x_2-x_2^*)] \\ \leq & [a_1-b_{11}(x_1+x_1^*)-b_{12}x_2-b_{13}x_3+b_{21}x_2^*+b_{31}x_3^*]|x_1-x_1^*| \\ &+[-a_2+b_{21}x_1-b_{22}(x_2+x_2^*)-b_{23}x_3^*+b_{12}x_1^*+b_{32}x_3]|x_2-x_2^*| \\ &+[-a_3+b_{31}x_1-b_{33}(x_3+x_3^*)-b_{32}x_2^*+b_{13}x_1^*+b_{23}x_2]|x_3-x_3^*| \\ \leq & [a_1-2b_{11}m_1^\epsilon-b_{12}m_2^\epsilon-b_{13}m_3^\epsilon+b_{21}M_2^\epsilon+b_{31}M_3^\epsilon]|x_1-x_1^*| \\ &+[-a_2+b_{21}M_1^\epsilon-2b_{22}m_2^\epsilon-b_{23}m_3^\epsilon+b_{12}M_1^\epsilon+b_{32}M_3^\epsilon]|x_2-x_2^*| \\ &+[-a_3+b_{31}M_1^\epsilon-2b_{33}m_3^\epsilon-b_{32}m_2^\epsilon+b_{13}M_1^\epsilon+b_{23}M_2^\epsilon]|x_3-x_3^*| \\ =&\sum_{i=1}^3 B_i^\epsilon(t) |x_i-x_i^*|. \end{align*} From the above arguments, there exists a positive constant $\mu>0$ such that \begin{equation} \label{8} D^+V(t)\leq -\mu \sum_{i=1}^3 |x_i(t)-x_i^*(t)| \quad \text{for all } t\geq T_1. \end{equation} Integrating both sides of \eqref{8} from $T_1$ to $t$, we obtain $$ V(t)+\mu \int_{T_1}^t \Big[\sum_{i=1}^3 |x_i(t)-x_i^*(t)|\Big]dt \leq V(T_1)<+\infty, t\geq T_1. $$ Then $$ \int_{T_1}^t \Big[\sum_{i=1}^3 |x_i(t)-x_i^*(t)|\Big]dt \leq \frac{1}{\mu} V(T_1)<+\infty, \quad t\geq T_1. $$ Hence, $\sum_{i=1}^3 |x_i(t)-x_i^*(t)| \in L^1([T_1, +\infty))$. On the other hand, the ultimate boundedness of $x_i$ and $x_i^*$ imply that both $x_i$ and $x_i^* (i\geq 1)$ have bounded derivatives for $t \geq T_1$. As a consequence $ \sum_{i=1}^3 |x_i(t)-x_i^*(t)|$ is uniformly continuous on $[T_1, +\infty)$. By Lemma \ref{lem.2} we have $$ \lim_{t \to \infty} \sum_{i=1}^3 |x_i(t)-x_i^*(t)|=0 $$ which completes the proof. \end{proof} \section{The model with constant interaction coefficients} In this section, we assume that the coefficients $b_{ij}$, $1\leq i,j\leq 3$ in system \eqref{1} are positive constants and the limit \[ M[a_i]=\lim_{T \to \infty} \frac{1}{T} \int_{t_0}^{t_0+T} a_i(t)dt \] exists uniformly with respect to $t_0$ in $(-\infty, \infty)$. First, we consider a predator-prey system \begin{equation} \label{10} \begin{gathered} x_1'(t)=x_1(t)[ a_1(t)-b_{11} x_1(t)-b_{12} x_2(t)],\\ x_2'(t)=x_2(t)[-a_2(t)+b_{21} x_1(t)-b_{22}x_2(t)]. \end{gathered} \end{equation} Put $Z_i(T)=\frac{1}{T} \int_{t_0}^{t_0+T} z_i(t)dt$. We have the following theorem. \begin{theorem} \label{thm4.1} Assume that $b_{11}b_{12}a_2^l+b_{11}b_{22}a_1^l-b_{12}b_{21}a_1^u>0$. Then $\inf_{t\geq t_0} x_1(t)$ $>0$. Furthermore, \begin{itemize} \item[(i)] If $M[a_2]<\frac{b_{21}}{b_{11}} M[a_1]$ then $\inf_{t\geq t_0} x_2(t)>0$ and $$ \lim_{T\to \infty} X_1(T)=\frac{b_{22}M[a_1]+b_{12}M[a_2]}{b_{12}b_{21} +b_{11}b_{22}},\quad \lim_{T\to \infty} X_2(T)=\frac{b_{21}M[a_1]-b_{11}M[a_2]} {b_{12}b_{21}+b_{11}b_{22}}. $$ \item[ii)] If $M[a_2]>\frac{b_{21}}{b_{11}} M[a_1]$ then $$ \lim_{T\to \infty} X_1(T)=\frac{M[a_1]}{b_{11}},\quad \lim_{T\to \infty} X_2(T)=0. $$ \end{itemize} \end{theorem} \begin{proof} The proof for the first statement is similar to that of Theorem \ref{thm3.1}. Let $\epsilon>0$ be a sufficiently small constant. From the comparison theorem and $x_1'(t)\leq x_1(t)[a_1^u-b_{11}x_1(t)]$, it is easy to see that $\limsup_{t\to \infty} x_1(t) \leq \frac{a_1^u}{b_{11}}$. Then there exists $T_1>t_0$ such that $x_1(t)< P_1^\epsilon=\frac{a_1^u}{b_{11}}+\epsilon$ for all $t\geq T_1$. Thus \begin{equation} \label{4.2} x_2'(t)0$ such that $-a_2^l+b_{21}P_1^\epsilon<0$. From \eqref{4.2}, it follows that $\lim_{t\to \infty} x_2(t)=0$. Therefore, there exists $T_2>T_1$ such that $a_1(t)-b_{12}x_2(t)>\frac{1}{2} a_1^l$. It follows from the first equation of the system \eqref{10} that $$ x_1'(t)\geq x_1(t)\big[\frac{1}{2}a_1^l-b_{11} x_1(t)\big] \quad \text{for } t\geq T_2. $$ Using the comparison theorem, we have $\liminf_{t\to \infty} x_1(t)\geq a_1^l/ 2b_{11}$. {\bf Case 2.} $-a_2^l+b_{21}P_1^0\geq 0$. It follows from \eqref{4.2} that $\limsup_{t\to\infty} x_2(t)\leq P_2^\epsilon=\frac{-a_2^l+b_{21}P_1^\epsilon}{b_{22}}$. Then, we can choose a sufficiently small positive $\epsilon$ and $T_3>T_1$ such that $x_1(t)\leq P_1^\epsilon, x_2(t)\leq P_2^\epsilon$ for all $t\geq T_3$. From the first equation of the system \eqref{10}, we have $x_1'(t)\geq x_1(t)[a_1^l-b_{12}P_2^\epsilon-b_{11}x_1(t)] \text{ for } t\geq T_3$. Because of our assumption $b_{11}b_{12}a_2^l+b_{11}b_{22}a_1^l$ $-b_{12}b_{21}a_1^u>0$, there exists a sufficiently small positive $\epsilon$ such that $$ a_1^l-b_{12}P_2^\epsilon=\frac{b_{11}b_{12}a_2^l+b_{11} b_{22}a_1^l-b_{12}b_{21}a_1^u}{b_{11}b_{22}} -\epsilon \frac{b_{12}b_{21}}{b_{22}}>0. $$ Then $\liminf_{t\to \infty} x_1(t)>0$. The conclusions of two above cases implies that $\inf_{t\geq t_0} x_1(t)>0$. Then there exists $c_1>0$ such that \begin{equation} \label{11} c_1< x_1(t)0. $$ If, contrary to the assertion of the theorem, $\inf_{t\geq t_0} x_2(t)=0$, then there exists a sequence of numbers $\{s_n\}_1^\infty$ such that $s_n\geq t_0, s_n\to \infty$ as $n\to \infty$ and $x_2(s_n)\to 0$ as $n\to \infty$. Put $$ c=\frac{1}{2}\liminf_{T\to \infty} \frac{1}{T} \int_{t_0}^{t_0+T} x_2(t)dt. $$ Since $x_2(t)>c$ for arbitrarily large values of $t$ and since $s_n\to \infty$ and $x_2(s_n)\to 0$ as $n\to \infty$, there exist sequences $\{p_n\}_1^\infty, \{q_n\}_1^\infty$ and $\{\tau_n\}_1^\infty$ such that for all $n\geq 1, t_0 t_n^*-\tau_n\geq n \quad \text {for } n\geq 1. \end{equation} In fact, $x_2'(t)=x_2(t)[-a_2(t)+b_{21} x_1(t)-b_{22}x_2(t)]0$. Therefore, there exists $c_2>0$ such that \begin{equation} \label{4.6} c_20$, we have \begin{gather*} \frac{1}{T} \ln \frac{x_1(t_0+T)}{x_1(t_0)} =A_1(T) -b_{11}X_1(T)-b_{12} X_2(T), \\ \frac{1}{T} \ln \frac{x_2(t_0+T)}{x_2(t_0)} =-A_2(T) +b_{21}X_1(T)-b_{22} X_2(T). \end{gather*} Then \begin{equation} \label{4.7} \begin{gathered} X_1(T)=\frac{b_{22}[A_1(T)-\frac{1}{T} \ln \frac{x_1(t_0+T)}{x_1(t_0)}]+b_{12}[\frac{1}{T} \ln \frac{x_2(t_0+T)}{x_2(t_0)}+A_2(T)]}{b_{12}b_{21}+b_{11}b_{22}}\,,\\ X_2(T)=\frac{b_{21}[A_1(T)-\frac{1}{T} \ln \frac{x_1(t_0+T)}{x_1(t_0)}] -b_{11}[\frac{1}{T} \ln \frac{x_2(t_0+T)}{x_2(t_0)} +A_2(T)]}{b_{12}b_{21}+b_{11}b_{22}}\,. \end{gathered} \end{equation} It follows from \eqref{11} and \eqref{4.6} that $$ \lim_{T\to \infty} \frac{1}{T} \ln \frac{x_i(t_0+T)}{x_i(t_0)}=0 \quad (i=1,2). $$ Then \begin{gather*} \lim_{T\to \infty} X_1(T)=\frac{b_{22}M[a_1] +b_{12}M[a_2]}{b_{12}b_{21}+b_{11}b_{22}}, \\ \lim_{T\to \infty} X_2(T)=\frac{b_{21}M[a_1]-b_{11}M[a_2]}{b_{12} b_{21}+b_{11}b_{22}}\,. \end{gather*} To prove Part (ii), first, we show that $\lim_{t\to \infty} x_2(t)=0$. Assuming the contrary we can find $\delta>0$ and a sequence of numbers $\{T_n\}_1^\infty, T_n>0, T_n\to \infty (n\to \infty)$ such that $\delta0$. This implies that $\lim_{t\to \infty} x_2(t)=0$ and then $\lim_{T\to \infty} X_2(T)=0$. It follows from the first equation of \eqref{4.7} that $\lim_{T\to \infty} X_1(T)=\frac{M[a_1]}{b_{11}}$. \end{proof} Now, we consider the system \begin{equation}\label{4.8} \begin{gathered} x_1'(t)=x_1(t)[ a_1(t)-b_{11} x_1(t)-b_{12} x_2(t)-b_{13} x_3(t)],\\ x_2'(t)=x_2(t)[ -a_2(t)+b_{21}x_1(t)-b_{22} x_2(t)-b_{23}x_3(t)],\\ x_3'(t)=x_3(t)[ -a_3(t)+b_{31} x_1(t)-b_{32} x_2(t)-b_{33} x_3(t)]. \end{gathered} \end{equation} \begin{proposition} \label{thm4.2} If \begin{equation} \label{4.9} \begin{gathered} b_{11}b_{12}a_2^l+b_{11}b_{22}a_1^l-b_{12}b_{21}a_1^u>0,\\ M[a_2]<\frac{b_{21}}{b_{11}} M[a_1],\\ M[a_3]<\frac{(b_{31}b_{22}-b_{32}b_{21})M[a_1] +(b_{31}b_{12}+b_{11}b_{32})M[a_2]}{b_{12}b_{21}+b_{11}b_{22}}, \end{gathered} \end{equation} then $\limsup_{t\to \infty} x_3(t)>0$. \end{proposition} \begin{proof} We assume that $\lim_{t\to \infty} x_3(t)=0$. Then \begin{equation} \label{4.9.1} \lim_{T\to \infty} X_3(T)=0. \end{equation} Replacing $t_0$ by a larger number, if necessary, we may assume that $a_1(t)-b_{13} x_3(t)>0$ for $t\geq t_0-1$. We put, \begin{gather*} a_1^*(t)= \begin{cases} a_1(t)-b_{13} x_3(t), & t\geq t_0,\\ a_1(t)-(t-t_0+1) b_{13} x_3(t), &t_0-1 \leq t < t_0,\\ a_1(t), & t0, a_i^{*u}<\infty$ for $ i=1,2$. Moreover, since $\lim_{t\to \infty} x_3(t)=0$, the limit $$ M[a_i^*]=\lim_{T\to \infty} \frac{1}{T}\int_{t_*}^{t_*+T} a_i^*(t) dt =\lim_{T\to \infty}\frac{1}{T} \int_{t_*}^{t_*+T} a_i(t) dt=M[a_i] $$ exists uniformly with respect to $t_*\in \mathbb{R}$ and $i=1,2$. Then for $t\geq t_0$, $(x_1(t), x_2(t))$ is a solution of the following competitive system \begin{gather*} x_1'(t)=x_1(t)\big[ a_1^*(t)-b_{11} x_1(t)-b_{12} x_2(t)\big],\\ x_2'(t)=x_2(t)\big[ -a_2^*(t)-b_{21}x_1(t)-b_{22} x_2(t)\big]. \end{gather*} By condition \eqref{4.9} and Theorem \ref{thm4.1}, we have \begin{equation} \label{4.9.3} \begin{gathered} \lim_{T\to \infty} X_1(T)=\frac{b_{22}M[a_1] +b_{12}M[a_2]}{b_{12}b_{21}+b_{11}b_{22}},\\ \lim_{T\to \infty} X_2(T)=\frac{b_{21}M[a_1] -b_{11}M[a_2]}{b_{12}b_{21}+b_{11}b_{22}}. \end{gathered} \end{equation} From the third equation of the system \eqref{4.8} we have $$ \frac{1}{T}\ln\big[\frac{x_3(t_0+T)}{x_3(t_0)}\big] =-A_3(T)+b_{31}X_1(T)-b_{32}X_2(T)-b_{33}X_3(T). $$ Then $-A_3(T)+b_{31}X_1(T)-b_{32}X_2(T)-b_{33}X_3(T)<0$ for $T$ sufficiently large. Letting $T\to \infty$ and using \eqref{4.9.1} and \eqref{4.9.3} we obtain $$ -M[a_3]+\frac{(b_{31}b_{22}-b_{32}b_{21})M[a_1]+(b_{12}b_{31} +b_{11}b_{32})M[a_2]}{b_{12}b_{21}+b_{11}b_{22}}\leq 0, $$ which contradicts \eqref{4.9}. This proves the proposition. \end{proof} \begin{proposition} \label{thm4.3} If the following conditions hold \begin{equation} \label{4.10} \begin{gathered} b_{11}b_{13}a_3^l+b_{11}b_{33}a_1^l-b_{13}b_{31}a_1^u>0,\\ M[a_3]<\frac{b_{31}}{b_{11}} M[a_1],\\ M[a_3]<\frac{(b_{31}b_{33}-b_{23}b_{31})M[a_1] +(b_{31}b_{13}+b_{11}b_{23})M[a_3]}{b_{13}b_{31}+b_{11}b_{33}} \end{gathered} \end{equation} then $\limsup_{t\to \infty} x_2(t)>0$. \end{proposition} The proof of the above proposition is similar to that of Proposition \ref {thm4.2}, and it is omitted. \begin{theorem} \label{thm4.5} If conditions \eqref{4.9} and \eqref{4.10} hold, then system \eqref{4.8} is persistent. \end{theorem} \begin{proof} From Propositions \ref{thm4.2} and \ref{thm4.3}, we have \begin{equation} \label{4.11} \limsup_{t\to \infty} x_i(t)>0, \quad i= 2,3. \end{equation} Now, we show that $\limsup_{t\to \infty} x_1(t)>0$. Assume the contrary, then there exist $t_1>t_0$ and two positive numbers $b_2, b_3$ such that $$ -a_i+b_{i1} x_1(t)<-b_i, \quad \text{for all } t\geq t_1, i=2, 3. $$ Then for $i=2, 3$ and $t\geq t_1$, $x_i'(t)\leq x_i(t)[-b_i-b_{ii} x_i(t)]$. By the comparison theorem, it follows that $\lim_{t\to \infty} x_i(t)=0$ which contradicts \eqref{4.11}. The proof is complete. \end{proof} \section{The model with the constant intrinsic growth rates} In this section, we consider system \eqref{1} under the condition $a_i, b_{ij}, 1\leq i,j \leq 3$ are constants, then \eqref{1} becomes \begin{equation}\label{5.1} \begin{gathered} x_1'(t)=x_1(t)[ a_1-b_{11} x_1(t)-b_{12} x_2(t)-b_{13} x_3(t)],\\ x_2'(t)=x_2(t)[-a_2+b_{21}x_1(t)-b_{22} x_2(t)-b_{23}x_3(t)],\\ x_3'(t)=x_3(t)[ -a_3+b_{31} x_1(t)-b_{32} x_2(t)-b_{33} x_3(t)]. \end{gathered} \end{equation} Put $$ x_1^*=\frac{a_1b_{22}+a_2b_{12}}{b_{11}b_{22}+b_{12}b_{21}},\quad x_2^*=\frac{a_1b_{21}-a_2b_{11}}{b_{11}b_{22}+b_{12}b_{21}}. $$ \begin{theorem} \label{thm4.6} If \[ a_2<\frac{b_{21}}{b_{11}} a_1\quad\text{and}\quad -a_3+b_{31}x_1^*-b_{32}x_2^*<0, \] then the stationary solution $(x_1^*, x_2^*, 0)$ of \eqref{5.1} is locally asymptotically stable. It means that if $(x_1(t), x_2(t), x_3(t))$ is a solution of \eqref{5.1} such that $(x_1(t_0),x_2(t_0))$ is close to $(x_1^*,x_2^*)$ and $x_3(t_0)$ is sufficiently small and positive, then $\lim_{t\to \infty} x_1(t)=x_1^*$, $\lim_{t\to \infty} x_2(t)=x_2^*$, $\lim_{t\to \infty} x_3(t)=0$. \end{theorem} \begin{proof} It is easy to see that $x_1^*>0, x_2^*>0$ and $(x_1^*, x_2^*, 0)$ is a stationary solution of system \eqref{5.1}. Put \begin{gather*} f_1(x_1,x_2,x_3)=x_1(a_1-b_{11} x_1-b_{12} x_2-b_{13} x_3), \\ f_2(x_1,x_2,x_3)=x_2(-a_2+b_{21} x_1-b_{22} x_2-b_{23} x_3), \\ f_3(x_1,x_2,x_3)=x_3(-a_3+b_{31} x_1-b_{32} x_2-b_{33} x_3), \end{gather*} then system \eqref{5.1} becomes $x_i'=f_i(x_1,x_2,x_3)$ and $f_i(x_1^*, x_2^*, 0)=0$, $i\geq1$. Consider \[ A= \begin{bmatrix} \frac{\partial f_1}{\partial x_1}& \frac{\partial f_1}{\partial x_2} & \frac{\partial f_1}{\partial x_3}\\ \frac{\partial f_2}{\partial x_1}&\frac{\partial f_2}{\partial x_2}& \frac{\partial f_2}{\partial x_3}\\ \frac{\partial f_3}{\partial x_1}&\frac{\partial f_3}{\partial x_2}&\frac{\partial f_3}{\partial x_3} \end{bmatrix} (x_1^*, x_2^*, 0)= \begin{bmatrix} -b_{11} x_1^*& -b_{12} x_1^* & -b_{13} x_1^*\\ b_{21} x_2^*& -b_{22} x_2^*& -b_{23} x_2^*\\ 0& 0& -a_3+b_{31}x_1^*-b_{32}x_2^* \end{bmatrix}. \] Since \[ \det (A-\lambda I) =(-a_3+b_{31} x_1^*-b_{32} x_2^*-\lambda)\big[\lambda^2 +(b_{11}x_1^*+b_{22} x_2^*) \lambda +(b_{11} b_{22}+b_{12}b_{21})x_1^*x_2^*\big], \] it follows that all eigenvalues of $A$ are less than zero. Therefore, $(x_1^*, x_2^*, 0)$ is locally asymptotically stable. \end{proof} \subsection*{Acknowledgements} The authors would like to thank the anonymous referee for his/her suggestions. 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