Abstract
Crop host-pathogen interaction have been a main issue for decades, in particular for food security. In this paper, we focus on the modeling and long term behavior of soil-borne pathogens. We first develop a new compartmental temporal model, which exhibits bi-stable asymptotical dynamics. To investigate the long term behavior, we use LaSalle Invariance Principle to derive sufficient conditions for global asymptotic stability of the pathogen free equilibrium and monotone dynamical systems theory to provide sufficient conditions for permanence of the system. Finally, we develop a partially degenerate reaction diffusion system, providing a numerical exploration based on the results obtained for the temporal system. We show that a traveling wave solution may exist where the speed of the wave follows a power law.
Highlights
The global food supply is currently experiencing pressure from climate change and ever increasing demand
There has been an increase in research of botanical pathogens and the resulting diseases, with foliar pathogens being the focus of the majority of published work
We provided sufficient conditions for Pathogen Free Equilibrium (PFE) being globally asymptotically stable and for persistence of the pathogen, using two different approaches, LaSalle Invariance Principle approach and monotone system approach
Summary
The global food supply is currently experiencing pressure from climate change and ever increasing demand. To some extent this paper is motivated by an early work of Gilligan [6], [7], [8], where he proposed a SEIR type compartmental model for the propagation of a soil-borne plant disease. Assumed that if the population of free pathogen is large, the transmission rate from S to I depends solely on a constant β and the level of susceptible hosts present. This type of incidence is called saturation incidence and we use the specific form βF M +F.
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