Nonlinear Age-Structured Models of Polycyclic Population Dynamics with Density Dependent Death Rate
In this work the explicit recurrent algorithms are obtained for two different age-structured nonlinear models of polycyclic population dynamics with density-dependent death rates. This work continues the study of polycyclic population dynamicsВ considering the effect of nonlinear mortality (population growth feedback) and fertility. In the first model the nonlinear death rates is described by power function of number of individuals in population with arbitrary exponent n.In the second one death rate is considered as a power function of population density with arbitrary exponent n. The temporal and age dynamics of population density in each model is governed by semi-linear transport equations with non-local integral boundary condition.
The explicit recurrent formulae of travelling wave solution are derived with the compatibility conditions for the coefficients of system and initial values. These formulae allow us to prove the theorem of existence and uniqueness of continuous and smooth travelling wave solution. The explicit recurrent algorithm is used also for development of accurate numerical algorithm for simulation of polycyclic population dynamics with two models of nonlinear death rates. The numerical experiments allow us to study the low of convergence of travelling wave solutions with nested difference meshes at the different moment of time and convergence of solutions to the exact particular solution of the differential problem. We obtain and study the discontinuous, continuous and smooth travelling wave solutions for the different types of initial values of population densities. The asymptotically stable travelling wave solutions are obtained and studied for autonomous systems with different values of exponent in power functions of nonlinear death rates. We observe also the population outbreaks in the form of quasi-periodical travelling wave solutions for the oscillating death rate and birth modulus for both models of death rate. Theoretical and applied results obtained in this work can be effectively used in the practical researches of biological population dynamics.
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