Nonlinear character dependent models with constant time delay in population dynamics
This paper is devoted to a nonlinear size structured model of population dynamics. An additional feature of the model includes a time delay in the birth process. The model has the following form: \[ p_ t(m,t)+g(m,t)p(m,t)+[g_ m(m,t)+\theta (m,P)]p(m,t)=0,\quad m>m_ 1, \] \[ P(t)=\int^{\infty}_{m_ 1}p(m',t)dm',\quad g(m_ 1,t)p(m_ 1,t)=P_ E\int^{\infty}_{m_ 1}\lambda (m',P)p(m',t-\tau_ E)dm',\quad t>\tau_ E, \] \[ p(m,0)=p_ 0(m),\quad m>m_ 1,\quad p_ m(m,t)=0,\quad t<0. \] Here n is the size variable, p(m,t) is the density of the population at time t with respect to size m, g(m,t) is the size and time dependent growth rate of individuals, P(t) is the total population at time t, \(\theta\) (m,P(t)) and \(\lambda\) (m,m',p(t)) are the size and population dependent mortality and fertility rates, respectively, \(m_ 1\) is the birth size, \(\tau_ E\) is the egg-stage duration, \(P_ E\) is the probability of survivorship, and \(p_ 0(m)\) is the initial size distribution. The properties studied are the local and global existence of solutions, the uniqueness of solutions, and the existence and stability of nontrivial equilibrium solutions. The method of characteristics is used to convert the problem to a pair of coupled nonlinear functional differential equations. The results generalize the age structured case of \textit{M. E. Gurtin} and \textit{R. C. MacCamy} [Arch. Rat. Mech. Analysis 54, 281-300 (1974; Zbl 0286.92005)] to a size structured case with time delay.
nonlinear size structured model of population dynamics
existence and stability of nontrivial equilibrium solutions
coupled nonlinear functional differential equations
scientific article; zbMATH DE number 4043670
scientific article