## Periodicity in a nonlinear discrete predator-prey system with state dependent delays.(English)Zbl 1152.34367

Summary: With the help of a continuation theorem based on Gaines and Mawhin’s coincidence degree, easily verifiable criteria are established for the global existence of positive periodic solutions of the following nonlinear discrete state dependent delays predator-prey system
\left\{\begin{aligned} N_1(k+1) & = N_1(k) \exp \left[ b_1(k)-\sum^n_{i=1} a_i(k)(N_1(k-\tau_i(k,N_1(k),N_2(k))))^{\alpha_i}\right. \\ & \left.- \sum^m_{j=1} c_j(k)(N_2(k-\sigma_j(k,N_1(k),N_2(k))))^{\beta_j}\right],\\ N_2 (k+1) & = N_2 (k) \exp \left[ -b_2(k)+\sum^n_{i=1} d_i (k) (N_1(k-\rho_i(k,N_1(k),N_2(k))))^{\gamma_i}\right]\end{aligned}\right.
where $$a_i,c_j,d_i:Z\rightarrow R^{+}$$ are positive $$\omega$$-periodic, $$\omega$$ is a fixed positive integer. $$b_{1},b_{2}:Z\rightarrow R^{+}$$ are $$\omega$$-periodic and $$\sum^{\omega-1}_{k=0} b_i(k) > 0$$. $$\tau_i,\sigma_j,\rho_i:Z\times R\times R\rightarrow R$$ ($$i=1,2,\dots, n,j=1,2,\dots ,m)$$ are $$\omega$$-periodic with respect to their first arguments, respectively. $$\alpha_i,\beta_j,\gamma_i$$ ($$i=1,2,\dots ,n,j=1,2,\dots ,m)$$ are positive constants.

### MSC:

 34K13 Periodic solutions to functional-differential equations 34C25 Periodic solutions to ordinary differential equations 34K20 Stability theory of functional-differential equations 92D25 Population dynamics (general)
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### References:

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