Mathematical Modelling of Plant-Pathogen-Herbivore Interaction Incorporating Allee Effect and Harvesting

Plant-pathogen-herbivore model describe interaction between plants, pathogens and herbivores. Plants are invaded by pathogens and herbivores while the herbivores are harvested by natural enemies such as predators and human. On the other hand, the abundance of food does not guarantee exponential growth of species who reproduce sexually and plants governed by carrying capacity. Therefore, the Allee effect may be crucial for sustaining such species. In this paper, a model of plant-pathogen-herbivore interactions that takes Allee effect and harvesting into account was developed and analyzed. The stability analysis showed that the ratio intrinsic growth rate to the environmental carrying capacity of susceptible plants must be greater than certain threshold value to raise sufficient plant biomass to sustain other species. Numerical simulations shows that all species coexist when intrinsic growth rate of plants is greater than the harvesting rate and when conversion rate of what is eaten by herbivores to newborn ones is greater than that of their natural enemies. It also shows that in the absence of susceptible plants, herbivores migrates in search of food, while others deteriorate and dies out. Furthermore, regardless of the availability of susceptible plants, the herbivores population crashes to extinction if the herbivore population is less than the lower limit required to keep the herbivores existing in the ecosystem.

In the interest of conservation of all species and the environment, policy developers will greatly benefit from understanding the solutions to address clearing land for human settlement, human activities and herbivore or their natural enemies hunting. In addition, monitor species closely, especially those that reproduce sexually by establishing and maintaining the least number required to keep the species existing.