Dynamic Analysis Of Predator-Prey System With Time Delay Of Toxin

Predator-prey interaction is an extremely important interspecific interaction in eco-logical theory.At the same time,it plays an important role in revealing the ecological mechanism and explaining ecological phenomena.In the long evolutionary process of in-teraction between plants and herbivores,plants get various forms of defensive measures.Among them,the secretion of toxins is a common form of self-protection.However,We need to rely on mathematical models and detailed analysis to be able to obtain influences of the element on system dynamics.Recently,the toxin-determined functional response was formulated by Li et al..And the toxins are released by terrestrial plants.He verified the suitability of the functional response by using theoretical analysis and numerical anal-ysis.But,the dynamic change of the system with this functional response is still needed to be further analyzed.Based on this,we consider dynamical behaviors of the system including time delay,diffusion and multi-species.Three models are built in this paper:a differential equation model with time delay of toxin,a delayed differential equation model with toxin and diffusion,a three-dimensional model with time delay of toxin.According to the differential equation model with time delay of toxin,the properties of stability and Hopf bifurcation of the model are analyzed in detail by using the theory of functional differential equations.And the conditions of stability and bifurcation of the system are acquired.Results show that the time delay of toxin is destabilizing the system,and making the state of poisonous plants and herbivores to be changed from stable to cyclical.For the delayed differential equation model with toxin and diffusion,based on the Turing stability theory and the central manifold theory,we also obtain the conditions of Turing instability and Hopf bifurcation.It is found that the time delay of toxin and the diffusion of the herbivores can also cause periodic oscillations of the system.However,the effect of the simple diffusion on the stability of the system is negligible.For the three-dimensional model with time delay of toxin,which consists of poisonous plants,non-toxic plants and herbivores,the properties of stability and bifurcation of the model are analyzed by using the theory of functional differential equation.And the conditions of coexistence and persistence of the three species are also obtained.It is concluded that the time delay of toxin makes the poisonous plants have no advantage.And it is possible that the non-toxic plants can survive.