ESTIMATES OF SOLUTIONS OF IMPULSIVE PARABOLIC EQUATIONS AND APPLICATIONS TO THE POPULATION DYNAMICS

A theorem on estimates of solutions of impulsive parabolic equations by means of solutions of impulsive ordinary differential equations is proved. An application to the population dynamics is given.


Introduction
The impulsive differential equations can be successfully used for mathematical simulation of processes and phenomena which are subject to short-term perturbations during their evolution.The duration of the perturbations is negligible in comparison with the duration of the process considered, and they can be thought of as momentary.The theory of impulsive ordinary differential equations started with the pioneer paper of V. Mil'man and A. Myshkis [17] and it was an object of intensive investigations during the last three decades.Detailed bibliographical information can be found in the monographs [2], [8]- [10], [16].
The theory of impulsive partial differential equations (PDE) marked its beginning with the paper [14].The impulsive PDE provide natural framework for mathematical simulation of many processes and phenomena in theoretical physics, population dynamics, bio-technologies, chemistry, impulse technique and economics.We would like to note the applications of the impulsive PDE in the quantum mechanics.In 1992 it was introduced a model of impulsive moving mirror [19], [20] presented by the apparatus of the impulsive PDE.
It must be pointed out that this theory is an object of many lectures delivered at international meetings.Note the lectures of C. Y. Chan, L. Ke [12] delivered at the First International Conference on Dynamic Systems and Applications, 1993, Atlanta, USA and S. Ahmad, M. Rama Mohana Rao [1] and D. Bainov, E. Minchev [7] delivered at the Second International Conference on Dynamic Systems and Applications, 1995, Atlanta, USA.
In this paper we give estimates of the solutions of impulsive parabolic equations and consider their applications to a model in the population dynamics.The estimates obtained can be used successfully in the qualitative theory of the impulsive parabolic equations.These estimates can be applied for obtaining of sufficient conditions for stability of the solutions of the equations investigated as well as they are useful for the entire formulation of the fundamental theory of the impulsive parabolic equations.

Preliminary notes
First we propose two models describing processes in the population dynamics.
Let Ω ⊂ R n be a bounded domain with a smooth boundary ∂Ω and Ω = Ω ∪ ∂Ω.Suppose that are given numbers such that lim k→∞ t k = +∞.
We define

Impulsive single species model. Suppose that f
Consider the reaction-diffusion equation (see [18]) and the impulses at fixed moments ( 4) The initial-boundary value problem (IBVP) ( 1)-( 4) describes a single species population in a bounded environment.The function u(t, x) represents the population density at the point x ∈ Ω and time t ≥ 0, f 0 (u) is the specific growth rate of u, κ > 0 is the diffusion coefficient, c 0 ≥ 0 is a constant.Condition (4) describes instantaneous changes in the population density due to phenomena as: harvesting, disasters, immigration, emigration, etc. Particularly, the case g(t k , x, u(t − k , x)) < 0 corresponds to instantaneous harvesting of a plant population (c 0 = 0) at times t k , k = 1, 2, . . ., while the case g(t k , x, u(t − k , x)) > 0 describes heavy immigration of a human population (c 0 > 0).
Motivated by the above models we consider initial boundary value problem for impulsive nonlinear parabolic equations.
Suppose that M [n] be the class of all matrices A = [a ij ] 1≤i,j≤n with real entries.Let where there exist continuous partial derivatives u t (t, x), u x (t, x), u xx (t, x) for (t, x) ∈ Γ and u satisfies (11) on Γ, (ii) u satisfies ( 12)-( 14).

Definition 2. A function
We introduce the following assumption: H1.The function f is elliptic at Γ.

Theorem 1. Let the following conditions hold:
1. Assumption H1 is fulfilled.2. There exist functions respectively, where Then for any solution u of the IBVP ( 11)-( 14) we have that × Ω and there exists a positive integer m such that t m < T 0 < t m+1 .We prove that There exists ε 0 > 0 such that for 0 < ε < ε 0 the solution γ(•; ε) of the problem (23) Suppose (24) is not true.Then the set Z = {t ∈ [0, T 0 ]: there exists x ∈ Ω such that u(t, x) ≥ γ(t; ε)} is non-empty.Defining t = inf Z.It follows from (19) and ( 20) that t > 0 and there exists a point x ∈ Ω such that: There are two cases to be distinguished: Case 1. ( t, x) ∈ Γ.Then we have From H1 and (15) we obtain that 0 ≤ u t ( t, x) − γ ( t; ε) which is a contradiction.
Analogously we can prove that r(t) ≤ u(t, x) on E T0 .
Since T 0 > 0 was arbitrary we get the estimates (21).

Effective estimates of the population density in the impulsive single species model
Particular interest for the mathematical biology is the special case of IBVP ( 1)-( 4) when pf 0 (p) = p(a − bp), a > 0, b > 0 are constants and c 0 = 0. Then the IBVP (1)-( 4) describing single species model takes on the form In the case without impulsive perturbations we have the above inequality with L = 1 and β = 0.
(17)here exist functions r(t) and γ(t) which are minimal and maximal solutions of the problems(17)