Nonlinear dynamics and spatial nonhomogeneity in entomopathogenic fungi growth on insect pest

Abstract

Biological control is the beneficial application of natural enemies such as pathogens, predators and parasites in managing insects pests and their damage. Entomopathogenic fungi (EPF) have a crucial role in natural ecosystems and have been developed as an environmentally friendly alternative to the use and application of chemical insecticides against insect pests. However, the dynamics of the entomopathogenic fungi within the insect host is still not well understood; due to the complexity behind the interaction between EPF, insects and their living environment which is really fluctuating. To study the dynamics of this system, we subdivided our work in three main points: in the first point host pathogen model is using to describe the intra-host dynamics of entomopathogenic fungi growth inside its host. The model is coupled with a nonlinear dependence of the consumption of insect resources by the host, described by the Holling and Powell type II functional responses. In the second point, a stochastic demography model (often called individual based model) defining as a random variation originating from the discrete nature of individuals is proposed to minic the outbreak of EPF within insect’s pests population; the model includes stochastic character of events like birth, death, infection and migration. Finally, the modified complex Ginzburg Landau equation (mGLCE) is used to model and to investigate the horizontal transmission between infectious insects and susceptible one. These studies show that the behavior of such system is rich in dynamics. Because the EPF growth is related to the instability of the system, particular attention is given to the stability analysis in this study. In the first part, the stability of system around the steady states is conducted with- out taking the diffusion into account. When considering a small perturbation of the stable singular point due to nonlinear diffusion, the conditions for Turing instability occurrence are deduced. It is observed that the absence of the regeneration feature of insect resources prevents the occurrence of such phenomena. The long time evolution of our system enables us to observe both spot and stripe patterns. Moreover, when the diffusion of mycelia is slightly modulated by a weak periodic perturbation, the Floquet theory and numerical simulations allow us to derive the conditions in which diffusion driven instabilities can occur. In the second part of our study, the stability analysis shows that the system dy- namics is strongly affected by the contagion rate between infectious insects and the susceptible hosts, where as the bifurcation analysis lead to a transcritical bifurcation when the basic reproduction number is greater than one in local dynamics. When migrations of species are considered, Hopf-damped Turing behavior can occur for a threshold contagion rate. However, sensitivity analysis of the extinction probability shows that the persistence of EPF depends to the proportion of spores collected from insect cadavers as well as their ability to be reactivated and create new infect insects. Lastly, the Anderson-may model is modified by taking in account the migration of infectious host. The model is transformed to a modified complex Ginzburg Lan- dau equation (mGCLE) using the multiple scale method. The effect of environmental conditionS is modeled in the infectious rate and the modulation instability (MI)of the wave plane is investigated. The linear stability analysis allows observing two types of modulation instabilities: Diffusion-driven instability or Turing instability and Paramet- ric instability observed when environmental condition influence the infection rate. The Floquet theory used in the latter case shown parametric resonance via the exhibition of Arnold tongues. However, the increase of the proportion of insect which pass from latent to infectious class increases the gain of the nonlinear instability and induced irregular behavior of MI in the case of constant and periodic infection.