MATHEMATICAL MODELING OF PLANT, PATHOGEN AND HERBIVORE INTERACTION INCORPORATING ALLEE EFFECT AND HARVESTING

Abstract

In ecology, interactions between various species that live in a similar habitat are common. In the plant-pathogen-herbivore interactions, plants are invaded by pathogens and herbivores while the herbivores are harvested by natural enemies such as predators and human. Harvesting the species may affect the population densities of the harvested species and keeping harvested species is critical in the ecosystem. On the other hand, the abundance of food does not guarantee exponential growth of species who reproduce sexually and species governed by carrying capacity. Therefore, the Allee effect may be crucial for sustaining such species. Recent plant-pathogen-herbivore models have not taken the Allee effect and species harvesting into account. The main objective of this study was to formulate and analyze a mathematical model of plant-pathogen-herbivore interactions incorporating Allee effect and harvesting. To illustrate the interaction of the species, the model was formulated using a system of ordinary differential equations. The local stability analysis was investigated and numerical simulations were performed using MATLAB software. The stability analysis showed that the ratio of 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. It also shows that the intrinsic growth rate of plants must be greater than the harvesting rate of plant population for plants to get established. Given this circumstance, all species coexists. Numerical simulations show 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 die out. Furthermore, regardless of the availability of susceptible plants, the herbivores population goes to extinction if the herbivore population is less than the least number 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 human activities for example, clearing land for farming, settlement, infrastructure construction, burning charcoal, 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.