Mathematical Modeling in Biology

Abstract

Abstract : In this thesis, it develops mathematical models that describe one aspect of the biological world: the mutual relationships between plants, animals, and their environment. It constructs a small part of the biological universe by considering the interactions of just two species. The relationship may be one of mutual aid, competition for limited resources, or predation of one upon the other. The mathematical modeling of two-species interactions is as follows: dx dt = x(a + a₁x + a₂y) dy = y(bo + bx + b₂y) dt } (1) where x(t) and y(t) denote the respective populations at time t of two species that form a closed interacting community and a's and b's are constants. If a = a2 = b1 = b2 = 0, then a Malthusian model is obtained, which involves the simplest birth and death process of a species' ecosystem. On the other hand, if a2 = b1 = 0, then (1) models a pair of noninteracting species, each of which follows the generalized logistic law. The steady-state orbits, which correspond to the constant and the periodic solutions of (1), and which are portrayed in the population quadrant (first quadrant) by points and simple closed curves (the No Cycles Theorem), are studied. The results appear to be contradictory. Specific predator-prey model accounts for. It constructs a simple mathematical model of a predator-prey community. This model was developed independently by Lotka-Volterra in the 1920s, who established the system. dx dt = x(a - cy) dy = y(-k + 2x) dt are the prey and predator populations (2) where x and y, respectively, at time t and a, c, k, and A are positive constants. Throughout the study, it focuses attention on qualitative behavior, explicit solutions, and the resulting ecological interpretations. It found that the solution trajectories in the phase plane of (2) model are closed and represent a periodic motion Then, the effect of an indiscriminate harvester or poison that kills both predator and prey is determined, and it is found that the average for prey is (k + H₂)/2 and for predator is (a - H₁)/c, where H₁ and H₁ are positive constants, H₂ represents the effect of harvesting on the prey, and H₁ represents the effect of harvesting on the predator. Harvesting of this type has surprising consequences with important ecological implications (Volterra's Principle), and then a discussion of this implication is presented. It is concluded with a discussion of some limitations of this model since its formulation