Beretta, Edoardo

Kon, Ryusuke

Takeuchi, Yasuhiro

Kyoto University Research Information Repository

TitleNon-existence of periodic solutions in delayed Lotka-Volterrasystems (Mathematical Models in Functional Equations)Author(s)Beretta, Edoardo; Kon, Ryusuke; Takeuchi, YasuhiroCitation数理解析研究所講究録 (2000), 1128: 19-27Issue Date2000-01URL http://hdl.handle.net/2433/63644RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityNon-existence of periodic solutions in delayedLotka-Volterra systems$\mathrm{E}\mathrm{d}\mathrm{o}\mathrm{a}\mathrm{J}:\mathrm{d}_{0}$ Beretta $1*$, Ryusuke Kon( ) 2and Yasuhiro $\mathrm{T}$ euchi ( ) $2\mathrm{T}$1 Istituto di Biomatematica, Universit\‘a di Urbino,I-61029 Urbino, Italy2 Department of Systems Engineering, Faculty of Engineering,Shizuoka University, Hamamatsu 432, Japan1 IntroductionIn this paper we derive sufficient conditions for the non-existence of nonconstant periodicsolutions of Volterra differential equations with distributed delays where the delay kemelsare chosen among $\gamma$-functions or their suitable convex normalized combinations. Thereason of this choice for the kernels is that the Volterra delay differential equations canthus be transformed in an expanded system of ordinary differential equations by thestandard ”linear chain trick” method [1]. To this expanded o.d.e. Volterra system wecan apply the conditions, encoded by the logarithmic norm of some Jacobian relatedmatrix, that Li and Muldowney [2] have obtained for the nonexistence of (nontrivial)periodic solutions for autonomous ordinary differential equations in $\mathrm{R}^{N}$ , conditions thatgeneralize to the case $N>2$ the Bendixon and Dulac critera.*This paper is performed in the frame of the research project Cofin $99$” $\mathrm{A}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{y}\mathrm{s}\mathrm{i}\mathrm{s}$ of Complex Systemsin Population Biology”.\dagger Research partly supported by the Ministry of Educationj Science and Culture, Japan, under Grant09640256.1128 2000 19-27 192 General resultsThe Volterra delay differential systems with distributed delays can be written as$\{$$\dot{x}_{i^{=}}X_{i}(e_{i}+j\sum_{=1}a_{i}jx_{j}+\sum\gamma nj=1nij\int^{t}-\infty tf_{ij(-}u)x_{j(}u)du)$ ,$i\in \mathrm{N}=\{1,2, \ldots n\}\triangle)$(2.1)where for each $\gamma_{ij}\neq 0,$ $f_{ij}$ : $[0, +\infty)arrow \mathrm{R}$ are continuous nonnegative functions obtainedby convex combination$f_{ij}(u)= \sum_{k=1}^{p_{i}}c_{ij}^{(}jk\rangle f_{ij}(k)(u)$ , $c_{ij}^{(k)}\geq 0$ , $\sum_{k=1}^{p_{i}j}c_{i}^{(}j^{)}=1k$ (2.2)of functions which are solutions of linear differential equations with constant coefficients:$f_{ij}^{(k)}(u)= \frac{\alpha_{ij}^{k}}{(k-1)!}u^{k-1}\exp(-\alpha_{ij}u)$ , $\alpha_{ij}\in \mathrm{R}_{+}$ , $k\in\{1,2, \ldots , p_{ij}\}$ (2.3)and satisfy the normalized condition$\int_{0}^{+\infty}f_{ij}(u)du=1$ .We remind that the average time delay of (2.3) is $T=k/\alpha_{ij}$ . We refer to (2.3) as to a$\gamma$-distribution (or $\gamma$-function) of order $k$ . According to linear chain trick ([1]) we put$\{$$x_{ij}^{\mathrm{t}}(k)t):= \int_{-\infty}^{t}f_{ij}(k)(t-u)X_{j(u})du$, $k=1,$ $\ldots,p_{ij}$ ,$x_{ij}^{(0)}(t):=x_{j}(t)$ , $i,j\in \mathrm{N}$ , $\gamma_{ij}\neq 0$ .(2.4)Let ” $p$” the number of distinct functions $x_{ij}^{(k)}$ and $P=\{n+1, \ldots , n+p\}$ the set of alltheir indices. According to (2.4), system (2.1) is transformed in an expanded system of$,,n+p$” ordinary differential equations$\{$$\dot{x}_{i}=x_{i}(e_{i}+j\sum_{=1}a_{ij}X_{j}+\sum\gamma nnpij\sum cijij(k)X_{ij})(k)$, $i\in \mathrm{N}$$\dot{x}_{ij}=(k)(\alpha ijX_{i}-j\alpha_{ij}xk-1)i(k)jj=,1k=1,.,p_{ij}k=1..$ , $i,j\in \mathrm{N}$ : $\gamma_{i}j\neq 0$(2.5)where the last ” $p$” are linear differential equations with real constant coefficients. Theinitial conditions for (2.1) require the knowledge in the past of the nonnegative, continuousand bounded functions$x_{i}(u)=\varphi_{i}(u),$ $u\in(-\infty, 0]$ for all $i\in \mathrm{N}$ . (2.6)20The (2.6) provide the $\mathrm{i}.\mathrm{c}$ . for (2.5). In fact:$\{$$\dot{x}_{i}(\mathrm{o})=\varphi_{i}(0)$ , $i\in \mathrm{N}$ ,$x_{ij}^{(k)}(0)= \int_{-\infty}^{0}f_{ij}(k)(-u)\varphi j(u)du$ , $k=1,$ $\ldots,p_{ij}$ , $i,j\in \mathrm{N}$(2.7)$\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{s}\underline{\mathrm{i}}\mathrm{d}\mathrm{e}\mathrm{r}$ the general system of differential equations$\frac{dx}{dt}=F(x)$ (2.8)where $F(x)\in \mathrm{R}^{N},$ $x\vdash’ F(x)$ is $C^{1}$ in an open subset $D_{0}$ of $\mathrm{R}^{N}$ . Denote by $J=(\partial F/\partial x)$the Jacobian of (2.8) and by $\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{N}$ the eigenvalues of $(1/2)[(\partial F/\partial x)+$$(\partial F/\partial_{X})^{T}]$ . Denote by $J^{[2]}$ the $\cross$ matrix which is the second additive compoundmatrix associated to the Jacobian matrix $J([2])$ and remind that if $x\in \mathrm{R}^{N}$ then thecorresponding logarithmic norms of $J^{[2]}$ (that we denote by $\mu(J^{[2]})$ ) endowed by the vectornorms (i) $|x|_{1}=\Sigma_{i}|x_{i}|,$ $( \mathrm{i}\mathrm{i})|x|_{\infty}=\sup_{i}|x_{i}|$ and (iii) $|x|_{2}=(x^{\tau_{x)}1}/2$ respectively are:$(\mathrm{i}\mathrm{i})(\mathrm{i})$ $\mu_{\infty}\mu_{1}(J^{[}2])(J[2])$ $==$ $\sup_{\sup}1_{\frac{}{\partial x_{r}}+\frac{\frac{\partial F_{s}}{\partial F_{s}\partial x_{s}}}{\partial x_{s}}+\sum}^{\frac{\partial F_{r}}{\partial F_{r}\partial x_{r}}+}+j\neq rj\neq\sum,(s)r,S(|\frac{}{\partial x_{j}}|+|\frac{\partial F_{j}}{\partial F_{r}\partial x_{r}}|+||\frac{}{\partial x_{j}}|)\frac{\partial F_{j}}{\partial F_{s}\partial x_{s}}|.\cdot.\cdot 1\leq r<\mathit{8}\leq N1\leq r<S\leq N1,’.$.(i\"u) $\mu_{2}(J^{[2]})$ $=$ $\lambda_{1}+\lambda_{2}$ ;where $\mu_{\infty}(J^{[2}])<0$ implies the diagonal dominance by row of the matrix $J^{[2]}$ and$\mu_{1}(J^{[2]})<0$ means its diagonal dominance by column. Then the following $\mathrm{h}\mathrm{o}\mathrm{l}\mathrm{d}_{\mathrm{S}}[2]$ :Theorem 2.1 If $\Omega\subset \mathrm{R}^{N}$ is a compact global attractor of (2.8) on which $\mu(J^{[2]})<0$for some logarithmic norm then in $\Omega$ there is no simple closed rectifiable curve which isinvariant with respect to (2.8).3 2-dimensional Volterra systems with 2 delaysNow let us consider $n$-dimensional Volterra delay differential systems with distributeddelays expressed by (2.1) with delay kernels (2.2) and (2.3). The systems can be expressedas (2.5) by using $p$ new variables (2.4) and become $(n+p)$-demensional o.d.e.. TheirJacobian has a size $(n+p)\cross(n+p)$ and its second additive compound, is $(_{2}^{n+p})\cross(_{2}^{n+\mathrm{p}})$ .Hence, in the following we restrict our systems with $n=2$ and $p\leq 2$ , that is, we consider21-dimensional Volterra systems with at most 2 delays, whose kernels are given by thefirst or second order $\gamma$-distributions ($k=1$ or 2 in$(2.3)$ ). Hereafter, for the simplicity ofnotation, we denote $x_{ij}^{(k)}$ as $x_{j}^{(k)}$ .Because of the symmetry of the systems, they are described as follows:$\bullet$ a system with one first order delay:$\{$$\dot{x}_{1}$ $=$ $x_{1}(e_{1}+a_{11}x_{1}+a_{12}x_{2}+\gamma x_{j}^{(1)})$$\dot{x_{2(1)}.}$$=$ $x_{2}(e_{2}+a21x1+a22^{X}2)$$x_{j}$$=$ $\alpha x_{j}-\alpha x_{j}^{(1)}$ $j=1$ or 2.(3.1)$\bullet$ a system with one-second order delay:. $\{$$\dot{x}_{1}$ $=$ $x_{1}(e_{1}+a_{11}x_{1}+a_{12}x_{2}+\gamma x_{j}^{(2)})$$\dot{x_{2(1)}.}$$=$$x_{2}(e_{2}+a_{21,(1}X_{1})+a22^{X}2)$$x_{j,(2)}$.$=$$\alpha x_{j,(1)}-\alpha x_{j}$$j$$=$$\alpha x_{j}$$-\alpha x_{j}^{(2)}$ , $j=1$ or 2.(3.2)$\bullet$ a system with $\mathrm{t}\mathrm{w}\{\succ \mathrm{f}\mathrm{i}\mathrm{r}\mathrm{S}\mathrm{t}$ order delays:$\{$$\dot{x}_{1}$$=x_{1}(e_{1}+a_{11}x_{1}+a_{12}x_{2}+\gamma_{1}x_{1}^{(1)}+\gamma_{2}x_{2}^{(1)})$$\dot{x_{2(1)}.}$$=$ $x_{2}(e_{2}+a_{21^{X_{1}}}+a22^{X}2)$$x_{1}$ $=$ $\alpha x_{1}-\alpha x_{1}$$\dot{x}_{2}^{(1)}$ $=\beta x_{2}-\beta x_{2}^{(1)}$(3.3)$\{$$\dot{x}_{1}$ $=x_{1}(e_{1}+a_{11}x_{1}+a_{12}x_{2}+\gamma_{1}x_{1}^{(1)})$$\dot{x_{2(1)}.}$$=x_{2}(e_{2}+a21X_{1}+(1)a_{222}X+\gamma_{2}x_{2}^{()})1$$x_{1}$ $=$ $\alpha x_{1}-\alpha x_{1}$$\dot{x}_{2}^{(1)}$ $=\beta x_{2}-\beta x_{2}^{(1)}$(3.4)$\{$$\dot{x}_{1}$ $=x_{1}(e_{1}+a_{11}x_{1}+a_{12}x_{2}+\gamma_{1}x_{2}^{()})1$$\dot{x_{2(1)}.}$$=x_{2}(e_{2}+a_{211}X+a22x2+\gamma_{21}X)(1)$$x_{1}$ $=$ $\alpha x_{1}-\alpha x_{1}$$\dot{x}_{2}^{(1)}$ $=\beta x_{2}-\beta x_{2}^{(1)}$(3.5)$\{$$\dot{x}_{1}$ $=x_{1}(e_{1}+a_{11}x_{1}+a_{12}x_{2}+\gamma_{1}x_{1}^{(1)})$$\dot{x_{2(1)}.}$$=x_{2}(e_{2}+a_{2,(1}1x1+a22x)2+\gamma_{21}\tilde{X})(1)$$x_{1,(1)}$.$=$ $\alpha x_{1}-\alpha X_{1}$$\tilde{x}_{1}$$=\beta x_{1}-\beta_{\tilde{X}_{1}}(1)$ .(3.6)We will distinguish between two systems in (3.1) as $(3.1)_{j}$ for $j=1,2$ . Similarly we definesystem $(3.2)_{j}$ for $j=1,2$ . For all systems, we always assume that $a_{ii}<0,$ $e_{i}\neq 0(i=1,2)$22and $\alpha,\beta>0$ . The first assumptions imply self-crowding effects biologically and the lastcomes from (2.3).First, we consider the boundedness and ’partial permanence’ of the solutions to sys-tems $(3.1)_{j}-(3.6)$ . Note that $\mathrm{R}_{+}^{3}$ or $\mathrm{R}_{+}^{4}$ is positive invariant for each system.Theorem 3.1 Suppose that$(a)$ for $(\mathit{3}.\mathit{1})_{1},\cdot$ one of the following is satisfied(a-l) $a_{12}a_{21}<0$ and $a_{11}+\gamma<0$(a-2) $a_{12}\leq 0,$ $a_{21}\leq 0$ and $a_{11}+\gamma<0$(a-3) $a_{11}a_{22}>a_{12}a_{21}$ and $\gamma<0$ :$(b)$ for $(\mathit{3}.\mathit{1})_{2_{f}}$. one of the folowing is satisfied(b-l) $a_{12}a_{21}<0$ and $a_{11}a_{22}>-\gamma^{2}a_{21}/(4a_{12})$(b-2) $a_{12}\leq 0$ and $a_{21}\leq 0$(b-3) $a_{11}a_{22}>a_{21}a_{21}$ and $\gamma\leq 0$ :$(c)$ for $(\mathit{3}.\mathit{2})_{1},\cdot$ one of the following is $\mathit{8}ati_{\mathit{8}}fied$(c-l) $a_{12}a_{21}<0$ and $a_{11}+|\gamma|<0$(c-2) $a_{12}\leq 0,$ $a_{21}\leq 0$ and $a_{11}+|\gamma|<0$(c-3) $a_{11}a_{22}>|a_{12}||a_{21}|_{f}a_{11}+|a_{12}|<0$ and $\gamma\leq 0$ :$(d)$ for $(\mathit{3}.\mathit{2})_{2}j$ one of the following is satisfied(d-l) $-a_{11}>|a_{12}|+|\gamma|and-a_{22}>|a_{21}|$(d-2) the same as (c-2)(d-3) the $\mathit{8}ame$ as (c-3):$(e)$ for $(\mathit{3}.\mathit{3})_{f}$. one of the following is satisfied(e-l) $-a_{11}>|a_{12}|+|\gamma_{1}|+|\gamma_{2}|and-a_{22}>|a_{21}|$23(e-2) $a_{12}\leq 0,$ $a_{21}\leq 0and-a_{11}>|\gamma_{1}|+|\gamma_{2}|$(e-3) $a_{12}a_{22}>|a_{12}||a_{21}|,$ $-a_{11}>|a_{12}|,$ $\gamma_{1}\leq 0$ and $\gamma_{2}\leq 0$ :$(f)$ for (3.4) or (3.5) or $(\mathit{3}.\theta)_{f}$. one of the following is satisfied(f-l) $a_{12}\leq 0_{f}a_{21}\leq 0_{f}-a_{11}>|\gamma_{1}|and-a_{22}>|\gamma_{2}|$(f-2) the same as (e-3).Then the solutions of $(\mathit{3}.\mathit{1})_{j^{-}}(\mathit{3}.\sigma)$ are bounded for any $\alpha>0$ and $\beta>0$ .Theorem 3.2 Suppose that the solutions of $(\mathit{3}.\mathit{1})_{j^{-}}.(\mathit{3}.\theta)$ are bounded and at least one of$e_{i}(i=1,2)$ is positive. Consider the solution $x(t)$ starting in $\mathrm{R}_{+}^{3}$ (system $(\mathit{3}.\mathit{1})_{j}$) or in$\mathrm{R}_{+}^{4}$ (system $(\mathit{3}.\mathit{2})_{j^{-}}(\mathit{3}.\theta)\mathit{1}$ . Choose a sufficiently large number $T>0$ and a sufficiently$\mathit{8}mall$ number $\epsilon>0$ and define sets$\Omega_{j}^{3}=\{X\in R_{+}^{\mathrm{s}}|x_{1}+x2>\in, xj(1)>0\}$ , $j=1,2$$\Omega^{4}=\{X\in R_{+}^{4}|x_{1}+X_{2}>\epsilon, xj(1)>0, j=1,2\}$$\overline{\Omega}^{4}=\{X\in R_{+}^{4}|x_{i}>\epsilon, x(i1)>0, i=1,2\}$$\tilde{\Omega}^{4}=\{X\in R_{+}^{4}|_{X_{1}}+X2>\mathcal{E}, x1(1)>0,\tilde{x}_{1}^{(1)}>0\}$ .(i) For $(\mathit{3}.\mathit{1})_{1_{f}}$ the solution stays in $\Omega_{1}^{3}$ for $t>T$, if $\gamma\leq 0or-a_{11}>\gamma>0_{f}$(ii) For $(\mathit{3}.\mathit{1})_{2}$ , the solution $stay\mathit{8}$ in $\Omega_{2}^{3}$ for $i>T,\cdot$(iii) $Suppo\mathit{8}ethat-a_{1}1>|\gamma|$ . Then for $(\mathit{3}.\mathit{2})_{1f}$ the solution stays in $\overline{\Omega}^{4}$ for $t>T_{f}$ if$e_{2}>a_{21}e_{1}/(a_{11}+\gamma)$ when $e_{1}>0$or $e_{1}>a_{12}e_{2}/a_{22}$ when $e_{2}>0$ ; (3.7)(iv) For $(\mathit{3}.\mathit{2})_{2}$ , the solution stays in $\overline{\Omega}^{4}$ for $t>T$ , if$e_{2}>a_{21}e_{1}/a_{11}$ when $e_{1}>0$or $e_{1}>e_{2}(a_{12}+\gamma)/a_{22}$ when $e_{2}>0$ ; (3.8)(v) For (3.3), the solution stays in $\Omega^{4}$ for $t>T,$ $if-a_{11}>|\gamma_{1}|_{f}$.24(vi) For $(\mathit{3}.\mathit{4})_{f}$ the solution stays in $\Omega^{4}$ for $t>T$, if$-a_{ii}>|\gamma_{i}|(i=1,2)$ ; (3.9)(vii) For $(\mathit{3}.\mathit{5})_{f}$ the solution stays in $\Omega^{4}$ for $t>\tau_{i}$(viii) For $(\mathit{3}.\theta)$, the solution $stay\mathit{8}$ in $\tilde{\Omega}^{4}$ for $t>T_{y}if-a_{11}>|\gamma_{1}|$ .4 Non-existence of periodic solutionsLet us apply Li-Muldowney’s criteria (Theorem 2.1) for the non-existence of periodicsolutions of systems $(3.1)_{j^{-}}(3.6)(j=1,2)$ . The Jacobian matrix of $(3.1)_{1}$ becomes$J=(e_{1}+2a_{111}X+a_{1}2^{X_{2}}+a21x_{2}\alpha\gamma x_{1}^{(}1)$$e_{2}+a_{211}a_{X+2a_{2}}12\mathrm{o}x12^{X_{2}}$ $\gamma x_{1}-\alpha 0)$ .The logarithmic norm $\mu_{1}$ endowed by the norm $|x|_{1}$ of the second additive compoundmatrix $J^{[2]}$ associated to $J$ is negative in $\mathrm{R}_{+0}^{3}$ if and only if the supremums of the followingfunctions satisfy$(e_{1}+2a_{11}x_{1}+a_{12}x_{2}+\gamma x_{1}^{(1)})+(e_{2}+a_{21}x_{1}+2a_{22^{X_{2}}})+\alpha<0$$(e_{1}+2a_{11}x_{1}+a_{12}x_{2}+\gamma x_{1}^{(1)})-\alpha+|a_{21}|X_{2}<0$$(e_{2}+a_{21}x_{1}+2a_{22^{X_{2}}})-\alpha+|a_{12}|x_{1}+|\gamma|x_{1}<0$,in $\mathrm{R}_{+0}^{3}$ . From the second and third inequalities, we have $a_{12}+|a_{21}|\leq 0$ and $a_{21}+|a_{12}|+$$|\gamma|\leq 0$ as necessary conditions for $\mu_{1}<0$ in $\mathrm{R}_{+0}^{3}$ . These two conditions hold true onlyfor $\gamma=0$ , which gives us a Lotka-Volterra system without a delay term. This shows thatthe direct application of Li-Muldowney’s method does not work for $(3.1)_{1}$ .Now let us transform $(3.1)_{1}$ by change of variables$\dot{y}_{1}=(e_{1}+a_{1\underline{1}}e^{\lambda y}11+a12e^{\lambda_{2}}y2+\gamma x^{(1}1))/\lambda_{1}$$\dot{y_{2(1)}.}=(e2+a_{21}e^{\lambda_{1}}y1+a_{2}2e^{\lambda_{2}}y2)/\lambda_{2}$ (4.1)$x_{1}$$=\alpha e^{\lambda_{1}y1}-\alpha x1(1)$where new variables $y_{i}(i=1,2)$ are defined by $y_{i}=(\log x_{i})/\lambda_{i}$ , for some positive constants$\lambda_{i}$ chosen later. The Jacobian matrix of (4.1) is$J_{1}^{1}=$ .25The logarithmic norm $\mu_{1}(J_{1}^{1[})2]$ is negative in $\mathrm{R}^{3}$ (note that it must be negative in $\mathrm{R}^{3}$ ,not in $\mathrm{R}_{+0}^{3}$ , because of change of variables) if and only if the folowing is satisfied in$\mathrm{R}^{3}$$\sup\{a_{11}e^{\lambda_{1y1}}+a22e^{\lambda_{2}\lambda_{1y}}+y2\alpha\lambda_{1}e\}1<0$$\sup\{a_{11}e^{\lambda_{1y1}}-\alpha+\lambda_{1}|a21|e^{\lambda}1y1/\lambda_{2}\}<0$$\sup\{a_{22}e\lambda_{2y2}-\alpha+\lambda 2|a_{1}2|e\lambda 2y_{2}/\lambda_{1}+|\gamma|/\lambda_{1}\}<0$. (4.2)Suppose that for sufficiently small $\epsilon>0$ and large $T>0$ , the following is satisfied by thesolution $y(t)=(y_{1}(t), y2(t),$ $x_{1}^{(1)}(t))$ of (4.1)$y(t)\in\Omega_{1y}^{3}=\{y\in R^{3}|e^{\lambda_{1y_{1}}}+e^{\lambda_{2y_{2}}}>\epsilon, x_{1}^{(1)}>0\}$ for $t>T$. (4.3)Under the assumption (4.3), the conditon given in Theorem 2.1 is ensured if$a_{11}+\alpha\lambda_{1}<0$ , $a_{11}+\lambda_{1}|a21|/\lambda_{2}\leq 0$ ,$a_{22}+\lambda_{2}|a12|/\lambda_{1}\leq 0$ , $-\alpha+|\gamma|/\lambda_{1}<0$ .The above is equivalent to$- \frac{|a_{21}|}{a_{11}}\leq\frac{\lambda_{2}}{\lambda_{1}}\leq-\frac{a_{22}}{|a_{12}|}$ , $\frac{|\gamma|}{\lambda_{1}}<\alpha<-\frac{a_{11}}{\lambda_{1}}$ . (4.4)Suppose that $a_{11}a_{22}\geq|a_{12}||a_{21}|$ and $-a_{11}>|\gamma|$ . Then it is easy to check that we canchoose $\lambda_{i}>0(i=1,2)$ satisfying (4.4) for each $\alpha>0$ . Note that $\Omega_{1y}^{3}$ corresponds to$\Omega_{1}^{3}$ defined in Section 3 and (4.3) is equivalent that the solution of $(3.1)_{1}$ stays in $\Omega_{1}^{3}$ for$t>T$ . For the last property, a sufficient condition is given in Theorem 3.2 (i). Thisproves the following Theorem 4.1 (i):Theorem 4.1 Suppose that the solutions of $(\mathit{3}.\mathit{1})_{j^{-}}(\mathit{3}.\theta)$ are bounded and at least one of$e_{i}(i=1,2)$ is positive. Then each system has no periodic solutions for any $\alpha>0$ and$\beta>0$ if the following conditions are satisfied:(i) For $(\mathit{3}.\mathit{1})_{1}$ ,$a_{11}a_{22}\geq|a_{1}2||a_{2}1|$ , $-a_{11}>|\gamma|$ ; (4.5)(ii) For $(\mathit{3}.\mathit{1})_{2}$ ,$a_{11}a_{22}\geq|a_{12}||a_{21}|$ , $a_{11}a_{22}>|a_{21}||\gamma|$ ; (4.6)26(iii) For $(\mathit{3}.\mathit{2})_{1f}(\mathit{3}.\mathit{1}\mathit{1})$ and$a_{22}(|\gamma|+a11)>|a_{12}||a_{2}1|$ ; (4.7)(iv) For $(\mathit{3}.\mathit{2})_{2},$ $(\mathit{3}.\mathit{1}\mathit{2})$ and$a_{11}a_{22}>|a_{21}|(|\gamma|+|a_{1}2|)$ ; (4.8)(v) For $(\mathit{3}.\mathit{3})_{f}$$a_{22}(|\gamma 1|+a_{11})>|a_{21}|(|\gamma 2|+|a_{1}2|)$ ; (4.9)(vi) For (3.4), (3.13) and$(a_{11}+|\gamma_{1}|)(a22+|\gamma_{2}|)>|a_{12}||a_{21}|$ ; (4.10)(vii) For $(\mathit{3}.\mathit{5})_{f}$$a_{11}a_{22}>(|\gamma_{1}|+|a_{1}2|)(|\gamma_{2}|+|a_{21}|))$. (4.11)(viii) For (3.6),$a_{22}(|\gamma_{1}|+a_{11})>|a_{12}||a_{21}|$ , $a_{11}a_{22}>|a_{12}|(|a_{21}|+|\gamma_{2}|)$ ; (4.12)and$|a_{21}|>|\gamma_{2}|$ ; (4.13)or (4. 12) and$-a_{11}|a_{21}|>(|a_{21}|+|\gamma_{2}|)|\gamma_{1}|$ , $2|\gamma_{1}|+a_{11}<0$ . (4.14)References[1] N. $\mathrm{M}\mathrm{a}\mathrm{c}\mathrm{D}_{\mathrm{o}\mathrm{n}}\mathrm{a}\mathrm{l}\mathrm{d}$, Time Lags in Biological Models, Springer, Berlin, 1978.[2] Y. Li, S. Muldowney, On Bendixon’s criterion, J. Diff. Eqn. 106, (1993) 27-39.27

Non-existence of periodic solutions in delayed Lotka-Volterra systems (Mathematical Models in Functional Equations)

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Non-existence of periodic solutions in delayed Lotka-Volterra systems (Mathematical Models in Functional Equations)

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