## The periodic predator-prey Lotka-Volterra model.(English)Zbl 0849.34026

The authors consider the Lotka-Volterra predator-prey model (1) $$u'(t) = \lambda l(t) u(t) - a(t) u^2 (t) - b(t) u(t) v(t)$$, $$v'(t) = \mu m(t) v(t) + c(t) u(t) v(t) - d(t) v^2(t)$$. Here $$l(t)$$, $$m(t)$$, $$a(t)$$, $$b(t)$$, $$c(t)$$ and $$d(t)$$ are continuous $$T$$-periodic functions such that $$a \geq 0$$, $$b \geq 0$$, $$c \geq 0$$, $$d \geq 0$$, $$a(t_1) > 0$$, $$d(t_2) > 0$$ for some $$t_1, t_2 \in \mathbb{R}$$, and ${1 \over T} \int^T_0 l(t)dt = 1, \quad {1 \over T} \int^T_0 m(t) dt = 1,$ the coefficients $$\lambda, \mu \in \mathbb{R}$$. The authors characterize the existence of coexistence states for the system (1) and analyze the dynamics of positive solutions of this system. Among other results they show that if some trivial or semi-trivial positive state is linearly stable, then it is globally asymptotically stable with respect to the positive solutions. In fact, the system possesses a coexistence state if, and only if, any of the semitrivial states is unstable. Some permanence and uniqueness results are also found. An example exhibiting a unique coexistence state that is unstable is given.

### MSC:

 34C25 Periodic solutions to ordinary differential equations 92D25 Population dynamics (general) 34D05 Asymptotic properties of solutions to ordinary differential equations