Multiple positive periodic solutions of a delayed discrete predator-prey system with type IV functional responses. (English) Zbl 1142.39015

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


39A11 Stability of difference equations (MSC2000)
92D25 Population dynamics (general)
39A20 Multiplicative and other generalized difference equations
Full Text: DOI


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