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Four positive periodic solutions to a Lotka-Volterra cooperative system with harvesting terms. (English) Zbl 1187.34050

Summary: We consider the following Lotka-Volterra cooperative system with harvesting terms:

$\left\{\begin{array}{c}{x}^{\text{'}}\left(t\right)=x\left(t\right)\left({a}_{1}\left(t\right)-{b}_{1}\left(t\right)x\left(t\right)+{c}_{1}\left(t\right)y\left(t\right)\right)-{h}_{1}\left(t\right),\hfill \\ {y}^{\text{'}}\left(t\right)=y\left(t\right){a}_{2}\left(t\right)-{b}_{2}\left(t\right)y\left(t\right)+{c}_{2}\left(t\right)x\left(t\right)\right)-{h}_{2}\left(t\right),\hfill \end{array}\right\\phantom{\rule{2.em}{0ex}}\left(1\right)$

where $x\left(t\right)$ and $y\left(t\right)$ denote the densities of two cooperative species, respectively: ${a}_{i}\left(t\right),{b}_{i}\left(t\right),{c}_{i}\left(t\right)$ and ${h}_{i}\left(t\right)$ $\left(i=1,2\right)$ are all positive continuous functions denoting the intrinsic growth rate, death rate, cooperative rate between the two species, harvesting rate, respectively.

We establish the existence of four positive periodic solutions of (1) by using the continuation theorem of coincidence degree.

##### MSC:
 34C25 Periodic solutions of ODE 92D25 Population dynamics (general) 47N20 Applications of operator theory to differential and integral equations
##### References:
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