Uniqueness of limit cycles in Gause-type models of predator-prey systems

Patterns in the effects of infectious diseases on population growth

“…This result illustrates that the periodic outbreaks for K slightly larger than K 1 are indeed huge. Dr. Jin Cheng-Fu [ 12] informs us that he can prove the uniqueness of the limit cycle for ~ ~ l by using a modified version of Zhang's theorem (see [ 14]). …”

Limit cycles in a Kolmogorov‐type model

“…4 It is impossible that the system (2.1) has two limit cycles. [xA+ [7]. Clearly, system (4.3) is a special case of (4.1) and hence if all the assumptions in this paper are satisfied, Theorem 3.7 is applicable.…”

“…In this paper, a Kolmogorov-type model, which includes the Gause-type model (Kuang and Freedman, 1988), the general predator-prey model (Huang 1988, Huang andMerrill 1989), and many other specialized models, is studied. The stability of equilibrium points, the existence and uniqueness of limit cycles in the model are proved.…”

Rich dynamics of a ratio-dependent one-prey two-predators model

“…This view is also plausible biologically (Cosner et al(1999)). When p(x) = αx/(m + x) and g(x/K) = r(1 − x/K), model (1.2) becomes the following well studied Michaelis-Menten type predator-prey system (see the references cited in Kuang and Freedman(1988)). where r, K, α, m, f, d are positive constants and x(t), y(t) represent the population density of prey and predator at time t respectively.…”