Estimates of solutions of impulsive parabolic equations and applications to the population dynamics

Publicacions Matem`atiques, Vol 40 (1996), 85–94. ESTIMATES OF SOLUTIONS OF IMPULSIVE PARABOLIC EQUATIONS AND APPLICATIONS TO THE POPULATION DYNAMICS Drumi Bainov and Emil Minchev Abstract 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 86 D. Bainov, E. Minchev 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. At the present time the theory of impulsive PDE undergoes rapid development [3]-[6], [11]-[15], [19], [20]. 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. 2. Preliminary notes First we propose two models describing processes in the population dynamics. Let Ω ⊂Rnbe a bounded domain with a smooth boundary ∂Ω and Ω=Ω∪∂Ω. Suppose that 0=t0<t 1<t 2<···<t k<··· are given numbers such that limk→∞ tk=+∞. We define E=(t, x)∈R1+n:t∈R+,x∈Ω,R+=[0,+∞), Γk=(t, x)∈E:t∈(tk,t k+1),x∈Ω,k=0,1,... ; Γ=∪∞ k=0Γk, Bk=(t, x)∈E:t∈(tk,t k+1),x∈∂Ω,k=0,1,... . Let Cimp[E,R] be the class of all functions u:E→Rsuch that: (i) The functions u|Γk∪Bk,k=0,1,..., are continuous. (ii) For each k,k=1,2,...,t=tk,there exists lim (s,q)→(t,x) s<t u(s, q)=u(t−,x),x∈Ω. (iii) For each k,k=0,1,...,t=tk, there exists lim (s,q)→(t,x) s>t u(s, q)=u(t+,x),x∈Ω, and u(t, x)=u(t+,x), x∈Ω. Estimates of solutions of impulsive parabolic equations 87 2.1. Impulsive single species model. Suppose that f0:R→R, u0: Ω →R,g:{tk}∞ k=1 ×Ω×R→Rare given functions. Consider the reaction-diffusion equation (see [18]) (1) ut(t, x)=κ∆u(t, x)+u(t, x)f0(u(t, x)) + c0,(t, x)∈Γ, subject to the initial condition (2) u(0,x)=u0(x),x∈Ω, the boundary condition (3) u(t, x)=0,(t, x)∈R+×∂Ω and the impulses at fixed moments (4) u(tk,x)=u(t− k,x)+g(tk,x,u(t− k,x)),x∈Ω,k=1,2,... . 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, f0(u) is the specific growth rate of u,κ>0 is the diffusion coefficient, c0≥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(tk,x,u(t− k,x)) <0 corresponds to instantaneous harvesting of a plant population (c0=0) at times tk,k=1,2,..., while the case g(tk,x,u(t− k,x)) >0 describes heavy immigration of a human population (c0>0). 2.2. Impulsive predator-prey system. Suppose that v(1) 0,v(2) 0: Ω→R,g(1),g(2) :{tk}∞ k=1 ×Ω×R2→Rare given functions. Consider a system describing predator-prey interaction v(1) t=∆v(1) +v(1)(˜a−v(1) −˜cv(2))inΓ,(5) v(2) t=∆v(2) +v(2)(−˜ d+˜ev(1) −v(2))inΓ,(6) v(1)(0,x)=v(1) 0(x),v (2)(0,x)=v(2) 0(x),x∈Ω,(7) v(1)(t, x)=v(2)(t, x)=0,(t, x)∈R+×∂Ω,(8) v(1)(tk,x)=v(1)(t− k,x)+g(1)(tk,x,v(1)(t− k,x),v(2)(t− k,x)),(9) v(2)(tk,x)=v(2)(t− k,x)+g(2)(tk,x,v(1)(t− k,x),v(2)(t− k,x)),(10) x∈Ω,k=1,2,... . 88 D. Bainov, E. Minchev In the IBVP (5)-(10) v(1)(t, x) denotes the population density of a prey and v(2)(t, x) that of a predator; ˜a,˜c,˜ d,˜eare positive constants. Conditions (9) and (10) represent instantaneous changes in the population density of the prey and the predator, respectively. For example, the case when g(1) ≡0 and g(2) <0 describes killing of the predators by hunters at the moments tk,k=1,2,.... Other possible situation is g(1) >0, g(2) ≡0 which corresponds to heavy immigration of the prey due to human interference. 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=[aij ]1≤i,j≤nwith real entries. Let f:Γ×R×Rn×M[n]→R,ϕ:R+×∂Ω→R, u0: Ω →R,g:{tk}∞ k=1 ×Ω×R→Rare given functions. Consider the initial-boundary value problem ut(t, x)=f(t, x, u(t, x),u x(t, x),u xx(t, x)),(t, x)∈Γ,(11) u(0,x)=u0(x),x∈Ω,(12) u(t, x)=ϕ(t, x),(t, x)∈R+×∂Ω,(13) u(tk,x)=u(t− k,x)+g(tk,x,u(t− k,x)),x∈Ω,k=1,2,... ,(14) where ux=(ux1,... ,u xn), uxx =[uxixj]1≤i,j≤n. Definition 1. A function u:E→Ris a solution of the IBVP (11)-(14) if: (i) u∈Cimp[E,R], there exist continuous partial derivatives ut(t, x), ux(t, x), uxx(t, x) for (t, x)∈Γ and usatisfies (11) on Γ, (ii) usatisfies (12)-(14). Definition 2. A function f:Γ×R×Rn×M[n]→Ris said to be elliptic at Γ if for each point (t, x)∈Γ and any Q,S∈M[n] the quadratic form n  i,j=1 (Qij −Sij)λiλj≤0 for arbitrary vector λ∈Rnimplies f(t, x, u, P, Q)≤f(t, x, u, P, S) for fixed (t, x, u, P)∈Γ×R×Rn. We introduce the following assumption: H1. The function fis elliptic at Γ. Estimates of solutions of impulsive parabolic equations 89 3. Main results Theorem 1. Let the following conditions hold: 1. Assumption H1 is fulfilled. 2. There exist functions f1,f2∈C((R+\{tk}∞ k=1)×R,R)such that (15) f1(t, p)≤f(t, x, p, 0,0) ≤f2(t, p) for (t, x)∈Γ,p∈R. 3. There exist functions g1,g2∈C({tk}∞ k=1 ×R,R)such that (16) g1(tk,p)≤g(tk,x,p)≤g2(tk,p), x∈Ω,p∈R,k=1,2,.... 4. The functions p+g1(tk,p)and p+g2(tk,p)are nondecreasing on Rfor each k,k=1,2,.... 5. There exist functions r(t)and γ(t)which are minimal and maximal solutions of the problems (17) r(t)=f1(t, r(t)),t=tk, r(0) = r0, r(tk)=r(t− k)+g1(tk,r(t− k)),k=1,2,... and (18) γ(t)=f2(t, γ(t)),t=tk, γ(0) = γ0, γ(tk)=γ(t− k)+g2(tk,γ(t− k)),k=1,2,... , respectively, where r0≤u0(x)≤γ0,x∈Ω,(19) r(t)≤ϕ(t, x)≤γ(t),(t, x)∈R+×∂Ω.(20) Then for any solution uof the IBVP (11)-(14) we have that (21) r(t)≤u(t, x)≤γ(t)on E. Proof: Let T0>0, ET0=[0,T 0]×Ω and there exists a positive integer msuch that tm<T 0<t m+1. We prove that (22) r(t)≤u(t, x)≤γ(t)onET0. 90 D. Bainov, E. Minchev There exists ε0>0 such that for 0 <ε<ε 0the solution γ(·;ε) of the problem (23) γ(t;ε)=f2(t, γ(t;ε)) + ε, t =tk, γ(0; ε)=γ(0) + ε, γ(tk;ε)=γ(t− k;ε)+g2(tk,γ(t− k;ε)) + ε, k =1,2,... ,m, is defined on [0,T 0] and limε→0γ(t;ε)=γ(t), uniformly on [0,T 0]. We prove that (24) u(t, x)<γ(t;ε)onET0. 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: (25) u(t, x)<γ(t;ε),(t, x)∈[0,˜ t)×Ω, u(˜ t, ˜x)=γ(˜ t;ε). There are two cases to be distinguished: Case 1. (˜ t, ˜x)∈Γ. Then we have ut(˜ t, ˜x)≥γ(˜ t;ε), ux(˜ t, ˜x)=0, n  i,j=1 uxixj(˜ t, ˜x)λiλj≤0,λ∈Rn. From H1 and (15) we obtain that 0≤ut(˜ t, ˜x)−γ(˜ t;ε) ≤f(˜ t, ˜x, u(˜ t, ˜x),0,0) −f2(˜ t, γ(˜ t;ε)) −ε<0, which is a contradiction. Case 2. ˜ t=tkfor some k,1≤k≤m. Then we have from (25) that (26) u(t− k,˜x)≤γ(t− k;ε), u(tk,˜x)=γ(tk;ε). From (16), (26) and Condition 4 of the theorem we conclude that 0=u(tk,˜x)−γ(tk;ε) =u(t− k,˜x)+g(tk,˜x, u(t− k,˜x)) −γ(t− k;ε)−g2(tk,γ(t− k;ε)) −ε ≤u(t− k,˜x)+g2(tk,u(t− k,˜x)) −γ(t− k;ε)−g2(tk,γ(t− k;ε)) −ε<0, Estimates of solutions of impulsive parabolic equations 91 which is a contradiction. Hence Zis empty and (24) follows. Since limε→0γ(t;ε)=γ(t) uniformly on [0,T 0] we conclude that u(t, x)≤γ(t)onET0. Analogously we can prove that r(t)≤u(t, x)onET0. Since T0>0 was arbitrary we get the estimates (21). Remark 1. Existence and uniqueness results for impulsive parabolic equations are considered in [5], [11] and [15]. 4. 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 pf0(p)=p(a−bp), a>0, b>0 are constants and c0= 0. Then the IBVP (1)-(4) describing single species model takes on the form ut(t, x)=κ∆u(t, x)+u(t, x)(a−bu(t, x)),(t, x)∈Γ,(27) u(0,x)=u0(x),x∈Ω,(28) u(t, x)=0,(t, x)∈R+×∂Ω,(29) u(tk,x)=u(t− k,x)+g(tk,x,u(t− k,x)),x∈Ω,k=1,2,... .(30) Suppose that g(tk,x,p)≤g2(tk,p)=βkp,βk>−1, x∈Ω, k= 1,2,..., and  s<tk≤t1 1+βk≥Le−β(t−s), where L>0, β>0 are constants, γ0= maxx∈Ωu0(x)>0. Let ube a solution of the IBVP (27)-(30). Then we consider the problem γ(t)=γ(t)(a−bγ(t)),t=tk, γ(0) = γ0, γ(tk)=γ(t− k)+βkγ(t− k),k=1,2,... . 92 D. Bainov, E. Minchev We substitute ρ(t)= 1 γ(t)and obtain the problem ρ(t)=−aρ(t)+b, t =tk, ρ(0) = ρ0=1 γ0 , ρ(tk)= 1 1+βk ρ(t− k),k=1,2,... . Then we have ρ(t)=ρ0 0<tk≤t1 1+βke−at + t  0 s<tk≤t1 1+βke−a(t−s)bds ≥Lρ0e−(a+β)t+Lb t  0 e−β(t−s)e−a(t−s)ds =Lρ0−Lb a+βe−(a+β)t+Lb a+β. Therefore γ(t)≤L1 γ0 −b a+βe−(a+β)t+Lb a+β−1 and by Theorem 1 we conclude that u(t, x)≤L1 γ0 −b a+βe−(a+β)t+Lb a+β−1 . In the case without impulsive perturbations we have the above inequality with L= 1 and β=0. Acknowledgements. The authors express their deep gratitude to the referee for his valuable advices and helpful suggestions. The present investigation was partially supported by the Bulgarian Ministry of Education, Science and Technologies under grant MM-422. References 1. S. Ahmad and M. Rama Mohana Rao, Stability of reaction-diffusion equations with impulsive effects, Proceedings of Dynamic Systems and Applications (to appear). Estimates of solutions of impulsive parabolic equations 93 2. D. Bainov and V. Covachev,“Impulsive Differential Equations with a Small Parameter,” World Scientific, Singapore, 1994. 3. D. Bainov, Z. Kamont, and E. Minchev, Difference methods for impulsive differential-functional equations, Appl. Numer. Math. 16 (1995), 401–416. 4. D. Bainov, Z. Kamont and E. Minchev, First order impulsive partial differential inequalities, Internat. J. Theoret. Phys. 33(6) (1994), 1341–1358. 5. D. Bainov, Z. Kamont and E. Minchev, Initial-boundary value problems for impulsive parabolic functional-differential equations, in “Boundary Value Problems for Functional Differential Equations,” ed. J. Henderson, World Scientific, Singapore, 1995, pp. 35–45. 6. D. Bainov, Z. Kamont and E. Minchev, Periodic boundary value problem for impulsive hyperbolic partial differential equations of first order, Appl. Math. Comput. 68 (1995), 95–104. 7. D. Bainov and E. Minchev, Trends in the theory of impulsive partial differential equations, Nonlinear World (to appear). 8. D. Bainov and P. Simeonov,“Systems with Impulse Effect: Stability, Theory and Applications,” Ellis Horwood, Chichester, 1989. 9. D. Bainov and P. Simeonov,“Theory of Impulsive Differential Equations: Asymptotic Properties of the Solutions,” World Scientific, Singapore, 1995. 10. D. Bainov and P. Simeonov,“Theory of Impulsive Differential Equations: Periodic Solutions and Applications,” Longman, Harlow, 1993. 11. L. Byszewski, Impulsive degenerate nonlinear parabolic functional-differential inequalities, J. Math. Anal. Appl. 164 (1992), 549–559. 12. C. Y. Chan and L. Ke, Remarks on impulsive quenching problems, Proceedings of Dynamic Systems and Applications 1(1994), 59–62. 13. C. Y. Chan, L. Ke and A. Vatsala, Impulsive quenching for reaction-diffusion equations, Nonlinear Anal. 22(11) (1994), 1323– 1328. 14. L. H. Erbe, H. I. Freedman, X. Z. Liu and J. H. Wu, Comparison principles for impulsive parabolic equations with applications to models of single species growth, J. Austral. Math. Soc. Ser. B 32 (1991), 382–400. 15. V. Gupta, Parabolic Equations with Impulse Effect: a Semigroup Approach, Ph. D. Thesis, Kanpur, India, 1994.