Evolution equations and diffusion operators for demographic dynamics

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 $L^p$ ($1⩽p⩽\mathrm{\infty }$) 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.

Keywords:

• Nonlinear evolution equation
• global classical solution
• stochastic model
• Ornstein–Uhlenbeck operator
• heat kernel

2000 Mathematics Subject Classifications:

• Primary: 92B05
• Secondary: 35Q91
• 65C20

Acknowledgments

Sheng-Ya Feng is partially supported by the National Natural Science Foundation of China [grant numbers 11501203 and 11426109], the China Postdoctoral Science Foundation [grant number 2016M600278] as well as the National Foundation of China Scholarship Council [grant number 201806745028]. Der-Chen Chang is partially supported by an NSF grant DMS-1408839 and a McDevitt Endowment Fund at Georgetown University.

Disclosure statement

No potential conflict of interest was reported by the authors.