Modeling Population Dynamics of Two Species with Volterra-Lotka Equations

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Population dynamics is a study of biological and environmental process. It is very important in the real world which describes the changes in population size and age composition. Many of the most interesting dynamics in the biological world have to do with interaction between species. Mathematical models which incorporate these interactions are required to stimulate these dynamics. So, in the 1920s, the Italian mathematician Vito Volterra (1860-1940) and Alfred Lotka developed the classical mathematical model of a predator-prey situation. This model called the Voltera-Lotka predatory-prey model (Penny, 2004). The equations of Voltera-Lotka were developed in papers by Lotka in 1925 and Volterra in 1926. The original system discovered by both Volterra and Lotka independently consisted of two entities. The model will involve two equations if we consider only two species. One which describes how the prey population changes and second which describes how the predator population changes. One species, the predators feeds on the other species, the prey which in turn feeds on same food item readily available in the environment. A standard example is a population of foxes and rabbits in woodland. The foxes (predators) eat rabbits (prey), while the rabbits eat certain vegetarian in the woodland. Other examples are sharks (predators) and food fish (prey), bass (predators) and sunfish (prey), ladybugs (predators) and aphids (prey), and beetles (predators) and scale insects (prey) (Penny, 2004). Beside the predator-prey situation, the other situation that occurs in population dynamics is competing species. These situations involve two interacting populations which population and compete with each other for the food in their same situation. Sometimes they will prey on one another but sometimes they will survive in peaceful. For example, two species of fish in a

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