Modeling of Biological Population Using Fuzzy Differential Equations: Fuzzy Predator-Prey Models and Numerical Solutions

Abstract

Abstract This thesis considers the application of fuzzy differential equations in modeling of predator and prey populations. When determining the initial populations of predator and prey, uncertainty can arise. We study a predator-prey model with different fuzzy initial populations using many cases of fuzzy numbers. The uncertainty can also arise when determining the birth and death rates of prey and predator, so we construct a fuzzy predator-prey model of fuzzy parameters. To the best of our knowledge, it is the first time to explore a fuzzy predator-prey model with functional response 𝑎𝑟𝑐𝑡𝑎𝑛(𝑎𝑥) and we study it with fuzzy initial populations and then with fuzzy parameters. We use generalized Hukuhara derivative and solve all models numerically by Runge-Kutta method. Simulations are made and graphical representations are also provided to show the evolution of both populations over time. At the end, we discuss the stability of the equilibrium points. From the simulations and graphs, we conclude that the fuzzy solution is not always better than the crisp solution biologically and sometimes they are unacceptable in fuzzy logic and some equilibrium points are unstable. We note that the solutions with triangular fuzzy numbers and shaped triangular fuzzy number are better than those with trapezoidal fuzzy numbers. As the initial populations of the prey and predator are closer to each other, the solution will be better since the lower and upper bounds are equal and positive. When we fuzzify the parameters of predator-prey model, we sometimes don’t get a good fuzzy solution. However, as the endpoints of fuzzy numbers are closer, the solution is periodic and the equilibrium points are stable