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

López-Gómez, Julián; Ortega, Rafael; Tineo, Antonio

Adv. Differ. Equ. 1, No. 3, 403-423 (1996).

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.

Reviewer: Chen Lan Sun (Beijing)

MSC:

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

Keywords:

Lotka-Volterra predator-prey model; coexistence states; positive solutions; permanence; uniqueness