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Abstract
In this paper, stochastic Hopf–Hopf bifurcation of the discrete coupling logistic system with symbiotic interaction is investigated. Firstly, orthogonal polynomial approximation of discrete random function in the Hilbert spaces is applied to reduce the discrete coupling logistic system with random parameter to the deterministic equivalent system. Then, it is concluded that Hopf–Hopf bifurcation exists in the equivalent deterministic system according to the principle of algebraic criteria. Numerical simulations show that the bifurcation critical value varies with the intensity of random parameter, and Hopf–Hopf bifurcation and period-doubling bifurcation behavior exist. In particular, Hopf–Hopf bifurcation can be drift with the change of random intensity, and frequency locking phenomenon occurs in the stochastic system.
Highlights
In 1798, Malthus proposed the population growth model which stated that the world population grew geometrically as the food and space resources became more abundant
We find that with the change of bifurcation parameter the phase trajectories of the deterministic system accord with the phase trajectories of two-species discrete coupling stochastic logistic system with symbiotic interaction, the bifurcation happens in both systems
According to the above numerical analysis, we discover that the critical value for Hopf–Hopf bifurcation in the two-species discrete coupling stochastic logistic system with symbiotic interaction is varying from random intensity
Summary
In 1798, Malthus proposed the population growth model which stated that the world population grew geometrically as the food and space resources became more abundant. The influence of a random parameter in the two-species coupled logistic system with symbiotic interaction on the Hopf–Hopf bifurcation is studied by orthogonal polynomial approximation [30,31,32,33,34,35,36]. 2, we transform the stochastic two-species coupling logistic system with symbiotic interaction with random parameter into its equivalent deterministic one by orthogonal polynomial approximation. According to the statistical characteristics and orthogonal polynomial approximation of discrete random function in the Hilbert spaces and the orthogonality of Charlier orthogonal polynomials, we can get the equivalent deterministic logistic equation. Theorem 2 (Existence of Hopf bifurcation) The two-species discrete coupling stochastic logistic system with symbiotic interaction undergoes the Hopf–Hopf bifurcation at the fixed point (0, 0) when the system parameter u goes by the critical value uc = –δ +.
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