Periodic solution for a predator-prey system with distributed delay. (English) Zbl 1102.34056

Lin, Guojian; Yuan, Rong

Math. Comput. Modelling 42, No. 9-10, 959-966 (2005).

Here, the two-dimensional system with distributed delay
\[ \dot{u}=\gamma u\Big(1-\frac{u}{K}\Big)- \frac{m}{y}\Big(\frac{vu}{a+u}\Big),\quad \dot{v}=v\bigg[m\Big(g^*\Big(\frac{u}{a+u}\Big)\Big)-d\bigg]\tag{1} \] is considered, where the parameters \(\gamma,K, m,y,a\) and \(d\) are positive constants, and
\[ g^*\Big(\frac{u}{a+u}\Big)(t)=\int_{-\infty}^t g(t-s)\frac{u(s)}{a+u(s)}\,ds,\quad t\,g(t)\in L^1((0,\infty);\mathbb R). \] Existence and stability of a periodic solution of system (1) are obtained.

Reviewer: R. G. Koplatadze (Tbilisi)

MSC:

34K13

Periodic solutions to functional-differential equations

Keywords:

periodic solution; distributed delay; predator-prey system; linear chain trick; normally hyperbolic; geometric singular perturbation theory

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