Abstract
This paper is motivated by the series of research papers that consider parasitoids’ external input upon the host–parasitoid interactions. We explore a class of host–parasitoid models with variable release and constant release of parasitoids. We assume that the host population has a constant rate of increase, but we do not assume any density dependence regulation other than parasitism acting on the host population. We compare the obtained results for constant stocking with the results for proportional stocking. We observe that under a specific condition, the release of a constant number of parasitoids can eventually drive the host population (pests) to extinction. There is always a boundary equilibrium where the host population extinct occurs, and the parasitoid population is stabilized at the constant stocking level. The constant and variable stocking can decrease the host population level in the unique interior equilibrium point; on the other hand, the parasitoid population level stays constant and does not depend on stocking. We prove the existence of Neimark–Sacker bifurcation and compute the approximation of the closed invariant curve. Then we consider a few host–parasitoid models with proportional and constant stocking, where we choose well-known probability functions of parasitism. By using the software package Mathematica we provide numerical simulations to support our study.
Highlights
Mathematical models of host–parasitoid interaction can exhibit an exciting and complex dynamics
In [17], by using the analytical approach we explore the global behavior and bifurcation in a class of host–parasitoid models when a constant number of the hosts are safe from parasitism: xn+1 = a + bxnf, yn+1 = cxn 1 – f, where a, b, c > 0, f is the proportion of hosts that are safe from parasitism and satisfies the following assumption: (H1) f ∈ C[0, ∞) ∩ C3(0, ∞), f (y) > 0, f (y) < 0, f (y) ≥ 0 for y > 0, and f (0) = 1, f (∞) = 0
3 Conclusion Successful biological control means that introduced natural enemies, often parasitoids, can control the pests and suppress them to the level where it can no longer cause economic damage [7, 24, 27, 28]
Summary
Mathematical models of host–parasitoid interaction can exhibit an exciting and complex dynamics. In system (4), it is assumed that a constant number of parasitoids is added per generation into host–parasitoid interaction to control the host population. We obtain that the constant stocking eliminates the equilibrium for which the host population survives and the parasitoid extinct. This equilibrium type exists in model (2) with no stocking (g(yn) ≡ 0) and model (3) with variable stocking, and in both systems, this equilibrium is semistable. A few examples of host– parasitoid models with proportional and constant stocking with well-known functions of probability of escaping parasitism are given in Sect. Lemma 3 For the boundary equilibrium of system (5), the following statements hold: (i) If –bxf (0) < 1, (x, 0) is locally asymptotically stable. We prove the existence and compute the direction of the Neimark–Sacker bifurcation
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