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Zbl 1142.39015
Zhang, Weipeng; Zhu, Deming; Bi, Ping
Multiple positive periodic solutions of a delayed discrete predator-prey system with type IV functional responses. (English)
[J] Appl. Math. Lett. 20, No. 10, 1031-1038 (2007). ISSN 0893-9659

The authors consider the following discrete predator-prey system with type IV functional responses and delays: $$\align x_1(k+ 1)&= x_1(k)\exp\Biggl[b_1(k)-a_1(k)x_1(k- \tau_1(k))-{c(k)x_2(k-\sigma(k))\over(x^2_1(k- \tau_2(k))/n)+ x_1(k-\tau_2(k))+ a}\Biggr],\\ x_2(k+1)&= x_2(k)\exp\Biggl[-b_2(k)+ {a_2(k)x_1(k-\tau_2(k))\over (x^2_1(k-\tau_2(k))/n)+ x_1(k- \tau_2(k))+ a}\Biggr] \endalign$$ (where for $i=1,2$, $b_i: Z\to\Bbb R$, $c,a_i: Z\to\Bbb R^+$, $\tau_i,\sigma: Z\to Z^+$ are all $\omega$ periodic, $n$ and $a$ are positive constants) for the initial condition $$\align x_1(-m)&\ge 0,\quad m=1,2,\dots,\max\{\tau_1(k), \tau_2(k),\sigma(k)\},\quad x(0)> 0.\\ x_2(-m)&\ge 0,\quad m= 1,2,\dots,\max\{\tau_1(k), \tau_2(k), \sigma(k)\},\quad y(0)> 0. \endalign$$ In the paper a theorem for the existence of positive periodic solutions of the system is given.
[Stefan Balint (Timişoara)]

MSC 2000:: *39A11 Stability of difference equations
92D25 Population dynamics
39A20 Generalized difference equations

Keywords: predator-prey systems; rational difference equations; type IV functional response; positive periodic solution; coincidence degree

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Zentralblatt MATH Berlin [Germany]