Abstract
ABSTRACT In this paper, we first review the related results of the evolution equations and operators in the parity theory of demographic evolution, and respectively establish differential equations and difference equations based on different initial states. In order to consider regional constraints, we modify the differential equation model to a stochastic Yule model and analyze the main features of the solution. Then, we add nonlinear terms to represent the limiting factors, transform the differential equation model into the initial value problem of nonlinear ordinary differential equations with parameters, and show the existence, stability and asymptotic behavior of global classical solutions in () spaces. Finally, we further study the demographic operator and the corresponding heat equation, and give the explicit structure of the singular geodesic of the associated Hamiltonian system and singular heat kernel of the operator.
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