In this paper periodic time-dependent Lotka-Volterra systems are
considered. It is shown that such a system has positive periodic solutions. It is
done without constructive conditions over the period and the parameters

Tsvetkov, D.

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Serdica Math. J. 22 (1996), 109-116A PERIODIC LOTKA-VOLTERRA SYSTEMD. P. TsvetkovCommunicated by V. PetkovAbstract. In this paper periodic time-dependent Lotka-Volterra systems areconsidered. It is shown that such a system has positive periodic solutions. It isdone without constructive conditions over the period and the parameters.1. The Periodic Lotka-Volterra System. Consider the Predator-Preymodel (see Volterra [1])N ′1 = (ε1 − γ1N2)N1N ′2 = (−ε2 + γ2N1)N2.(1)The functions N1 and N2 measure the sizes of the Prey and Predator populationsrespectively. The coefficients ε1, ε2, γ1, γ2 are assumed as nonnegative ω-periodicfunctions of time t. The period ω > 0 is arbitrary chosen and fixed. This periodicityassumption is natural; one may see for instance the work of J. Cushing [2] in whichis given a satisfactory justification on it. We still recount (due to [2]) some periodicfactors like seasonal effects of weather, food supply, mating habits, hunting or harvestingseasons, etc. Here one may add any unidirectional ω-periodic influence of anotherpredator over the prey.We will look for ω-periodic positive solutions for the conservative system (1)that corresponds to the nature of N1 and N2.1991 Mathematics Subject Classification: 34A25, 92B20Key words: periodic Lotka-Volterra system Predator-Prey110 D. P. TsvetkovThis work presents a result of existence. Notice that in the following theoremthere are no conditions on the period and there are no constructive conditions on theparameters of the system. Our result is obtained under weak assumptions. However,it is not explicit that makes the solutions difficult to any further examination.Theorem. Suppose that ε1, ε2, γ1, γ2 are nonnegative continuous ω-periodicfunctions and that each of them is not equal to zero identically. Then there exist ω-periodic solutions with N1(t) > 0 and N2(t) > 0 for t ∈ R. Moreover, these solutionssatisfy the inequalitiesminN1 ≥ e−ωmax ε1 maxN1, minN2 ≥ e−ωmax ε2 maxN2andmaxN1 ≤∫ ω0 ε2(s)ds∫ ω0 γ2(s)dseωmax ε1 , maxN2 ≤∫ ω0 ε1(s)ds∫ ω0 γ1(s)dseωmax ε2 .The present work is related to the mentioned paper of J. Cushing [2] whoconsidered the systemN ′1 = (b1 − c11N1 − c12N2)N1N ′2 = (−b2 + c21N1 − c22N2)N2and has proved existence theorems. It is done under the constructive conditionc11(t)c12(t) > 0 for all t that makes the addend c11N1 unremovable. Therefore, thereis no intersect between the results of [2] and the above Theorem.The framework of the present paper is closed to the papers of Z. Amine andR. Ortega [3] and R. Ortega and A. Tineo [4] in which the authors considered theLotka-Volterra systemu′ = (a(t)− b(t)u− c(t)v) uv′ = (d(t)± f(t)u− g(t)v) vunder the condition that the coefficients are strictly positive.In this connection notice the the paper of A. Tineo and C. Alvarez [5] in whichthe authors, due to K. Gopalsamy, studied the periodic solutions of competing systemsu′i = ui[bi −n∑j=1aijuj], 1 ≤ i ≤ n,(n ≥ 2) under the conditionsmin(bi) >∑j 6=imax(aij)min(ajj)max(bj), 1 ≤ i ≤ n,A Periodic Lotka-Volterra System 111that implies min(ajj) > 0. So there is no overlap between the approach of the works[2]–[5] and the approach of the present paper.Finally, notice the example of J. Kolesov and D. Shvitra in the book [6] (anactual system which includes delay effects) in which the self-existed oscillations in thePrey equation force the oscillations in the Predator equation.2. Examples.Fig. 1 Fig. 2 Fig. 3Let us investigate numerically the following systemN′1 = (sin2t− cos 22t N2)N1N′2 = (−cos2t+ sin 23t N1)N2.A pi-periodic solution is found near the initial data N1(0) = 0.9004 and N2(0) = 1.0728.The calculations give|N1(0)−N1(pi)| < 0.00001, |N2(0)−N2(pi)| < 0.00002.Its form is shown in Fig. 1. Repeat the same for the systemN′1 = (sin23t− 2cos 2t N2)N1N′2 = (−cos2t+ 3sin 2t N1)N2.A pi-periodic solution is found near the initial data N1(0) = 0.2646 and N2(0) = 0.4755.The calculations give|N1(0) −N1(pi)| < 0.0005, |N2(0)−N2(pi)| < 0.0009.Its form is shown in Fig. 2. Finally consider the systemN′1 = (sin23t− 2N2)N1N′2 = (−cos27t+ 3N1)N2.112 D. P. TsvetkovA pi-periodic solution is found near the initial data N1(0) = 0.166 and N2(0) = 0.252for which|N1(0)−N1(pi)| < 0.00006, |N2(0)−N2(pi)| < 0.00057.Its significant form is given in Fig. 3.3. Proof of the Theorem. Denoteε1def= maxtε1(t), ε2def= maxtε2(t).In view of our assumptions we have ε1 6= 0 and ε2 6= 0. Denoteu(t)def= ε1 − ε1(t), v(t)def= ε2 − ε2(t).Obviously u(t) ≥ 0, v(t) ≥ 0, t ∈ R. Now rewrite equations (1) in the form−N ′1(t) + ε1N1(t) = (u(t) + γ1N2(t))N1(t)N ′2(t) + ε2N2(t) = (v(t) + γ2N1(t))N2(t).(2)By the ω-periodic Green functionsG1(t, ω) =eε1teε1ω − 1, G2(t, ω) =e−ε2t1− e−ε2ω, t ∈ [0, ω),the problem for ω-periodic solutions of (2) is reduced to the problem for continuousω-periodic solutions of the following operator systemN1 =∫ ω0 G1(t− s, ω) (u(s) + γ1N2(s))N1(s)dsdef= X (N1, N2)N2 =∫ ω0 G2(t− s, ω) (v(s) + γ2N1(s))N2(s)dsdef= Y(N1, N2).Denote by C(ω) the space of the real continuous ω-periodic functions defined on thewhole axis. Let X be the Banach space C(ω)⊗ C(ω) with the conventional norm‖(N1, N2)‖X = maxt|N1(t)|+maxt|N2(t)|.It is not difficult to see that the operatorZdef= (X ,Y) : X → Xis completely continuous. Moreover Z is positive with respect to the coneKdef= {(N1, N2) ∈ X : N1(t) ≥ 0 and N2(t) ≥ 0; t ∈ R}i.e. Z : K → K. It can be shown that Z is positive with respect to the subconeK◦ ⊂ KK◦def= {mintN1(t) ≥ e−ε1ωmaxtN1(t) and mintN2(t) ≥ e−ε2ω maxtN2(t)}.A Periodic Lotka-Volterra System 113We pay more attention to this phenomenon in view of its importance. In factwe have Z : K → K◦ sincemintX (N1, N2) ≥minG1maxG1maxtX (N1, N2) = e−ε1ω maxtX (N1, N2)andmintY(N1, N2) ≥minG2maxG2maxtY(N1, N2) = e−ε2ωmaxtY(N1, N2).whenever (N1, N2) ∈ K. One can find similar estimates in M. Krasnosel’skii, E. Lifshicand A. Sobolev [7].The proof is based on the theory of completely continuous vector fields presentedby M. Krasnosel’skii and P. Zabrejko in [8]. The following proposition is extracted from[8] in a form convenient for us.Proposition [8]. Let Y be a real Banach space with a cone Q and L : Y → Ybe a completely continuous and positive (L : Q→ Q) with respect to Q operator. Thenthe following assertions are valid.i. Let L(0) = 0. Let also L be differentable at zero with a derivative L′(0) andthere is no y ∈ Q, y 6= 0, withy◦≤ L′(0)y.Then there exists ind(0, L;Q) = 1.ii. Let, for every sufficiently large R, there is no y ∈ Q with‖y‖Y = R and L(y)◦≤ y.Then there exists ind(∞, L;Q) = 0.iii. Let L(0) = 0 and let there exist ind(0, L;Q) 6= ind(∞, L;Q). Then L has anontrivial fixed point in Q.Here ind(·, L;Q) denotes the index of a point with respect to L and Q. Thesign◦≤ denotes the semiordering generated by Q.Of course, Z is differentable at zero with a derivativeZ ′(0)(N1, N2) =(∫ ω0G1(t− s, ω)u(s)N1(s)ds,∫ ω0G2(t− s, ω)v(s)N2(s)ds).Let us show that there is no nontrivial (N1, N2) ∈ K◦ such that the coordinate inequal-ity(N1(t), N2(t)) ≤ Z′(0)(N1, N2)(t), t ∈ R,holds. Otherwise there is (N˜1, N˜2), with the mentioned property, for which, withoutloss of generality, we assume N˜1 6≡0. Then integrating at [0, ω] we obtain(3)∫ ω0N˜1(s)ds ≤1ε1∫ ω0u(s)N˜1(s)ds,114 D. P. Tsvetkovwhich leads to the following contradiction∫ ω0(ε1 − u(s))N˜1(s)ds ≤ 0since, under the definition ofK◦, we have mint N˜1(t) > 0 and the nonnegative differenceε1 − u ≡ ε1 does not equal to zero identically. Thus point i of the cited propositionyieldsind(0,Z;K◦) = 1.Therefore, in accordance with point iii, for a proof of our theorem it is enough to showthat(4) ind(∞,Z;K◦) = 0which we are going to do.Let Z∗ be a positive, with respect to the cone K◦, operator defined as followsZ∗(N1, N2) = (X∗(N1, N2),Y∗(N1, N2)) ==(eε1ωω∫ ω0N1(s)ds+ 1,eε2ωω∫ ω0N2(s)ds + 1).At first we shall prove that, the completely continuous and positive with respect to K◦fields, I − Z and I − Z∗ are positive linear homotopic atDRdef= {(N1, N2) ∈ K◦ : ‖(N1, N2)‖X = 2R}where R is chosen arbitrary with(5) R > ω e(ε1+ε2)ω max(ε1∫ ω0 γ1(s)ds,ε2∫ ω0 γ2(s)ds).Otherwise, within the definitions (see [8]), there exist (N˜1, N˜2) ∈ DR and θ˜ ∈ [0, 1] forwhich(6)θ˜X (N˜1, N˜2) + (1− θ˜)X∗(N˜1, N˜2) = N˜1θ˜Y(N˜1, N˜2) + (1− θ˜)Y∗(N˜1, N˜2) = N˜2.Without loss of generality, we assume maxt N˜2(t) ≥ R. ThenmintN˜2(t) ≥ Re−ε2ω.At this point the first equality of (6) impliesmaxtN˜1(t) e−ε1ωRθ˜e−ε2ω∫ ω0G1(t− s, ω)γ1(s)ds+A Periodic Lotka-Volterra System 115+maxtN˜1(t) (1− θ˜) + (1− θ˜) ≤ maxtN˜1(t), t ∈ R.Integrating the last at [0, ω] we obtainmaxtN˜1(t) θ˜(Re−ε2ωe−ε1ωε1∫ ω0γ1(s)ds− ω)+ ω(1− θ˜) ≤ 0which, in view of (5), may hold if and only if the function N˜1 is equal to zero identicallyand θ˜ = 1. Then substituting the values found for N˜1 and θ˜ in the second equality of(6) we get ∫ ω0G2(t− s, ω)v(s)N˜2(s)ds = N˜2(t), t ∈ R.This leads to a contradiction in the same way as (3). Thus we prove the aforementionedhomotopy.At last we are going to show thatind(∞,Z∗;K◦) = 0which implies the validity of (4), since the homotopic fields have the same index. Herewe use point ii of our proposition. For this purpose it is enough to observe that thereis no (N˜1, N˜2) ∈ K◦ withX ∗(N˜1, N˜2)(t) =eε1ωω∫ ω0N˜1(s)ds + 1 ≤ N˜1(t), t ∈ R.Otherwise, after integrating at [0, ω], the last gives the impossible inequalityeε1ω∫ ω0N˜1(s)ds+ ω ≤∫ ω0N˜1(s)ds.Thus we prove that system (1) has nontrivial solutions.The proof of the second part follows from the definition of K◦ and from thefact that for every solution it holds∫ ω0ε1(s)ds−∫ ω0γ1(s)N2(s)ds =∫ ω0N ′1(s)N1(s)ds = 0and ∫ ω0ε2(s)ds −∫ ω0γ2(s)N1(s)ds =∫ ω0N ′2(s)N2(s)ds = 0.4. Notices. The most important detail in the proof was to obtain a propergrowth of Z at infinity (in order to find ind(∞,Z, ·)) that forces the introduction ofthe cone K◦.116 D. P. TsvetkovIt will be of certain interest to investigate the existence of positive almost-periodic solutions of (1) with positive almost-periodic coefficients. Perhaps this problemis much more difficult than the periodic one. In this case, together with the compactnessof the solution operator, we (possibly) lose the opportunity to use a convenient conelike K◦.REFERENCES[1] V. Volterra. Theory of Functionals and of Integral and Integro-DifferentialEquations. Dover Publications Inc., New York, 1958.[2] J. M. Cushing. Periodic time-dependent Predator-Prey systems. SIAM J. Appl.Math. 32 (1977), 82-95.[3] Z. Amine, R. Ortega. A periodic Prey-Predator system. J. Math. Anal. Appl.185 (1994), 477-489.[4] R. Ortega, A. Tineo. On the number of positive periodic solutions for planarcompeting Lotka-Volterra systems. J. Math. Anal. Appl. 193 (1995), 975-978.[5] A. Tineo, C. Alvarez. A different consideration about the globally asymptoti-cally stable solution of the periodic n-competing species problem. J. Math. Anal.Appl. 159 (1991), 44-50.[6] J. S. Kolesov, D. I. Shvitra. Self-Existed Oscillations in Systems with Delay.Mocslas, Vilnus, 1979, (in Russian).[7] M. A. Krasnosel’skii, E. A. Lifshic, A. V. Sobolev. Positive Linear Systems.Nauka, Moscow, 1985, (in Russian).[8] M. A. Krasnosel’skii, P. P. Zabrejko. Geometrical Methods of NonlinearAnalysis. Nauka, Moscow, 1975, (in Russian).Dimiter Petkov TsvetkovDepartment of Mathematics and InformaticsUniversity of Veliko Tarnovo5000 Veliko TarnovoBulgaria Received January 29, 1996

A Periodic Lotka-Volterra System

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A Periodic Lotka-Volterra System

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