Nonlinear age-structured models of polycyclic population dynamics with death rates as power functions with exponent n

摘要

This paper is devoted to the development of explicit recurrent algorithm and numerical study of properties of travelling wave solutions of two age-structured population dynamics models with nonlinear death rates and polycyclic reproduction condition. Death rate of first model is a power function with arbitrary exponent n of total number of individuals in population (integral characteristic of population density). In the second model death rate is a power function with arbitrary exponent n of population density. Both models are considered as the systems of initial-boundary value problems for semi-linear transport equations with non-local integral boundary condition. The explicit recurrent algorithm for the numerical solution of this system is derived with restrictions for the coefficients of equations and initial values which guarantee the existence of a unique local continuous and smooth travelling wave solution. Recurrent formulae allow us to build the accurate numerical algorithm and carry out the numerous simulations for the different scenarios of population dynamics with the set of parameterized algebraic functions. We carry out the numerical study of: (1) convergence of numerical solutions with nested meshes at the different moment of time and convergence of numerical solutions to the exact particular solution of system; (2) discontinuous, continuous and smooth travelling wave solutions for the different initial values of population densities; (3) asymptotically stable states for the different values of exponent n of power function of nonlinear death rates; (4) quasi-periodical behaviour of population density for the oscillating death rate and birth modulus. Results obtained in this study allow us to understand the particularities of different regimes of population dynamics and discover new properties of travelling wave solution of non-linear polycyclic age-structured model.