Generalized Volterra Equations Research Articles

Numerical modeling of dynamics of a population system with time delay

Mathematical models of interacting populations are often constructed as systems of differential equations, which describe how populations change with time. Below we study such a model connected to the nonlinear dynamics of a system of populations in presence of time delay. The consequence of the presence of the time delay is that the nonlinear dynamics of the studied system become more rich, eg, new orbits in the phase space of the system arise, which are dependent on the time‐delay parameters. In more detail, we introduce a time delay and generalize the model system of differential equations for the interaction of 3 populations based on generalized Volterra equations in which the growth rates and competition coefficients of populations depend on the number of members of all populations. Then we solve numerically the system with and without time delay. We use a modification of the method of Adams for the numerical solution of the system of model equations with time delay. By appropriate selection of the parameters and initial conditions, we show the impact of the delay time on the dynamics of the studied population system.

Positive invariance of closed sets for generalized Volterra equations

Positive invariance of closed sets for generalized Volterra equations

The stability of generalized Volterra equations

In this paper, a particular type of a system of generalized Volterra equations [1], whose solutions are assured to be nonnegative for arbitrary nonnegative initial values, is considered. The extended stability theorem of LaSalle is used for deriving conditions for a nonnegative equilibrium point to be stable with respect to a certain subset of the Euclidean space. The obtained stability theorem has a close relation with Lyapunov's stability condition for linear systems with constant coefficients and is generally less restrictive than conditions known so far.

<i>M</i>関数を用いた一般化ボルテラ方程式系の安定性

<i>M</i>関数を用いた一般化ボルテラ方程式系の安定性

Periodic solutions in n dimensions and Volterra equations

If a nonlinear autonomous n-dimensional system of ordinary differential equations has a bounded solution with a certain uniform stability property, this solution approaches an almost periodic solution with the same stability property. (More precisely, the almost periodic solution is in the set of ω-limit points of the given solution.) If the bounded solution has, in addition to the uniform stability property, an asymptotic stability property, then the solution approaches a periodic solution with the same stability properties. Practical (i.e., computable) sufficient conditions for boundedness of solutions are obtained. The results are applied to generalized Volterra equations.