On stability of cooperative and hereditary systems with a distributed delay
aa r X i v : . [ m a t h . D S ] J un On stability of cooperative and hereditary systemswith a distributed delay
Leonid Berezansky and Elena Braverman Dept. of Math, Ben-Gurion University of Negev, Beer-Sheva 84105, IsraelE-mail: [email protected] Dept. of Math & Stats, University of Calgary, 2500 University Dr. NW, Calgary,AB, Canada T2N 1N4E-mail: [email protected]
Abstract.
We consider a system dxdt = r ( t ) G ( x ) "Z th ( t ) f ( y ( s )) d s R ( t, s ) − x ( t ) ,dydt = r ( t ) G ( y ) "Z th ( t ) f ( x ( s )) d s R ( t, s ) − y ( t ) with increasing functions f and f , which has at most one positive equilibrium. Here the values of the functions r i , G i , f i are positive for positive arguments, the delays in the cooperative term canbe distributed and unbounded, both systems with concentrated delays and integro-differential systems are a particular case of the considered system. Analyzing therelation of the functions f and f , we obtain several possible scenarios of the globalbehaviour. They include the cases when all nontrivial positive solutions tend to thesame attractor which can be the positive equilibrium, the origin or infinity. Anotherpossibility is the dependency of asymptotics on the initial conditions: either solutionswith large enough initial values tend to the equilibrium, while others tend to zero,or solutions with small enough initial values tend to the equilibrium, while others in-finitely grow. In some sense solutions of the equation are intrinsically non-oscillatory:if both initial functions are less/greater than the equilibrium value, so is the solutionfor any positive time value. The paper continues the study of equations with monotoneproduction functions initiated in [Nonlinearity, 2013, 2833-2849]. AMS Subject Classification:
Keywords: cooperative systems of differential equations, distributed delay, globalattractivity, permanent solutions tability of cooperative systems with distributed delays
1. Introduction
The system of autonomous differential equations with constant delays in the productionterm dxdt = R ( y ( t − τ )) − a x ( t ) dydt = R ( x ( t − τ )) − a y ( t ) (1.1)was considered in [30], where R i : (0 , ∞ ) → (0 , ∞ ) are monotone increasing functions.It can describe a couple of populations, where the growth of each population isstimulated by the size of the other population and is suppressed by its own growth.Systems of differential equations describing different types of species, where the rateof change for each of them is positively influenced by all other populations but itself,are usually called cooperative . This is in contrast, for example, to competitive systems,where this influence is negative, and predator-prey systems, with different types ofinfluences. These systems can correspond to the cooperative types of species, or to thepatch environment, the growth in each patch is suppressed by overpopulation in itselfwhile stimulated by high density in adjacent patched, due, for example, to possibleimmigration. Another situation is hereditary systems where each variable describesa different developmental stage of the same species (e.g. eggs, larvae, juveniles, adultspecies capable of reproduction). In the case of system (1.1), x and y can be juvenile andadult counts, respectively. There is a competition within each group, as well as naturalmortality, and the mortality per capita rate is assumed to be population-independent.All the growth of juveniles is due to reproduction of adults, while maturation ofjuveniles contributes to adult numbers. There are delays in both recruitment processes(maturation delay for juveniles and reproduction time for adults). In line with the abovedescription, model (1.1) includes delay in the reproduction term only, and the mortalityis assumed to be proportional to the current population density.In the present paper, we consider systems of two equations where the growth ofeach of two variables is stimulated by high numbers in the other (due to cooperation,or inheriting part of it, or influx of offspring of the other population), and call themcooperative or hereditary systems. The delays of a positive impact can describe thetime required to translate nutritional benefits into body mass for the cooperation type.For hereditary systems, we have maturation and reproduction delays.System (1.1) includes the two-neuron bidirectional associative memory (BAM)model [15] x ′ ( t ) = − x ( t ) + af ( y ( t )) + I, y ′ = − y ( t ) + bg ( x ( t )) + J. (1.2)A simplified version of the delay system considered in [8] dxdt = c tanh( y ( t − τ )) − µ x ( t ) dydt = c tanh( x ( t − τ )) − µ y ( t ) (1.3) tability of cooperative systems with distributed delays dxdt = G ( x ( t )) [ R ( y ( t − τ )) − a x ( t )] dydt = G ( y ( t )) [ R ( x ( t − τ )) − a y ( t )] (1.4)includes a system of logistic equations with the delay in the production term; equationsof this type were described in [1]. Some particular non-delay systems of type (1.4)were studied in [29]. For example, the Lotka-Volterra cooperative model considered in[21, 27, 22], if the delayed mortality terms are omitted, has the form dxdt = x ( t ) [ r − a x ( t ) + b y ( t − τ )] dydt = y ( t ) [ r − a y ( t ) + b x ( t − τ )] . (1.5)Evidently (1.5) is a particular case of (1.4), and all the results of [30] are applicable to(1.5).The Hopfield neural network [13] x ′ i ( t ) = − b i ( x i ( t )) + n X j =1 c ij f j ( x j ( t )) + I i , t ≥ , i = 1 , . . . , n, (1.6)with n = 2 and c ii = 0, can be rewritten as (1.4) with τ = τ = 0, arbitrary a i > G i ( u ) = b i ( u ) / ( a i u ), R i ( u ) = c ij f j ( u ) /G i ( u ), j = i .The purpose of the present paper is to explore global asymptotic stability ofcooperative systems with a distributed delay, which include (1.1) and (1.4) as specialcases; in addition to being distributed, the delay can change with time. Distributeddelays describe a feasible fact that any interval for delay value has some probability,such models include equations with concentrated (either constant or variable) delays.Stability of equations and systems with distributed delays attracted recently muchattention, see, for example, [2, 3, 4, 6, 10, 11, 12, 17, 18, 19, 23, 24, 25, 26, 28, 31]for some recent results and their applications, also see references therein. The summaryof the results obtained by the beginning of 1990ies can be found in [16]. The methodsapplied to establish absolute convergence of the system either to the origin, or to theunique positive equilibrium, or to infinity, goes back to [5, 6] and was applied in [3, 4].In contrast to our earlier papers [5, 6, 3, 4], in the present paper we consider a system,not a single equations. Compared to all other previous work, the main differences areoutlined below. • We consider distributed delays of the most general type; as particular cases, theyinclude systems with variable concentrated delays, integral terms (in most papers,distributed delay is associated with these integral terms), their combination, andsome other models (for example, Cantor function as a distribution). Moreover, tability of cooperative systems with distributed delays • The delay distributions can be non-autonomous. If we describe these distributionsas a probability that a delay takes a greater than a given value, this corresponds totime-dependent delay. In applications, this allows to consider, for example, seasonalchanges in delay distributions. To some extent, we explore the most general systemwith a unique positive equilibrium, and justify global stability of this equilibrium,once delays are involved in those terms only which describe cross-influences. Thisis a generalization of the result in [30] for a system of two autonomous equationswith constant concentrated delays. To some extent, we have answered the questionwhen delays do not have any destabilizing effect on a non-autonomous system oftwo equations. • On the other hand, many of the previous papers on distributed delay describe muchmore complicated dynamics than absolute global stability established in the presentpaper. For example, delay dependence of stability properties was studied in [6],while possible multistability considered in [4]. However, the study of systems whichcan be destabilized by large enough delay are not in the framework of the presentpaper. Here we restrict ourselves to monotone increasing production functions,which can be treated as positive feedback in the delayed term.The paper is organized as follows. Section 2 contains existence, positivity andpermanence results for models with a distributed delay. Section 3 presents the globalstability theorem which is the main result of the present paper. Finally, Section 4considers applications and involves some discussion.
2. Positivity and Solution Bounds
In the present paper we consider the system with a distributed delay dxdt = r ( t ) (cid:20)Z th ( t ) f ( y ( s )) d s R ( t, s ) − x ( t ) (cid:21) dydt = r ( t ) (cid:20)Z th ( t ) f ( x ( s )) d s R ( t, s ) − y ( t ) (cid:21) (2.1)with the initial conditions x ( t ) = ϕ ( t ) , t ≤ , y ( t ) = ψ ( t ) , t ≤ , (2.2)where ϕ ( t ) and ψ ( t ) are initial functions. Definition 2.1
The pair of functions ( x ( t ) , y ( t )) is a solution of system (2.1) , (2.2) if it satisfies (2.1) for almost all t ≥ and (2.2) for t ≤ . System (2.1) will be investigated under some of the following assumptions: tability of cooperative systems with distributed delays (a1) f i : R + → R + = [0 , ∞ ), i = 1 , f i are strictly monotoneincreasing on R + ( f i ( x ) > f i ( y ) for x > y ≥
0) and f i ( x ) > x > i = 1 , (a2) The equation f − ( x ) = f ( x ) has exactly one positive solution K >
0, where f ( x ) > f − ( x ) for f (0) < x < K and f ( x ) < f − ( x ) for x > K ; (a3) h i : R + → R , i = 1 , h i ( t ) ≤ t ,lim t →∞ h i ( t ) = ∞ , i = 1 , (a4) R i ( t, · ), i = 1 , t , R i ( · , s ) arelocally integrable for any s , R i ( t, s ) = 0, s ≤ h i ( t ), R i ( t, t + ) = 1, r i ( t ) are Lebesguemeasurable essentially bounded on R + functions, r i ( t ) ≥ i = 1 ,
2; here u ( t + ) isthe right-side limit of function u at point t . (a5) Z ∞ r i ( s ) ds = ∞ , i = 1 , (a6) ϕ : ( −∞ , → R and ψ : ( −∞ , → R are continuous bounded functions, ϕ ( t ) ≥ ψ ( t ) ≥ t < ϕ (0) > ψ (0) > x ( t ) , y ( t )) = ( K, f ( K )).As particular cases, system (2.1) includes the model with variable delays dxdt = r ( t ) [ f ( y ( h ( t ))) − x ( t )] dydt = r ( t ) [ f ( x ( h ( t ))) − y ( t )] (2.3)where instead of (a4) we assume (b4) r i ( t ) are Lebesgue measurable essentially bounded on R + functions, r i ( t ) ≥ i = 1 , dxdt = r ( t ) (cid:20)Z th ( t ) K ( t, s ) f ( y ( s )) ds − x ( t ) (cid:21) dydt = r ( t ) (cid:20)Z th ( t ) K ( t, s ) f ( x ( s )) ds − y ( t ) (cid:21) (2.4)where instead of (a4) we consider the condition (c4) K i ( t, s ) : R + × R + → R + , i = 1 , t and s satisfying Z th i ( t ) K i ( t, s ) ds ≡ r i ( t ) are Lebesgue measurable essentially boundedon R + functions, r i ( t ) ≥ i = 1 , Definition 2.2
The solution ( x ( t ) , y ( t )) of (2.1) , (2.2) is permanent if there exist a , b and A , B , A ≥ a > , B ≥ b > , such that a ≤ x ( t ) ≤ A, b ≤ y ( t ) ≤ B, t ≥ . tability of cooperative systems with distributed delays , ∞ ). Theorem 2.3
Suppose (a1),(a3)-(a4),(a6) hold.1) A solution of (2.1) , (2.2) is positive in its maximal interval of existence [0 , d ) .2) If in addition (a2 ∗ ) there exists K > such that f ( x ) < f − ( x ) for x > K then there exists a positive solution of (2.1) , (2.2) for t ∈ [0 , ∞ ) . We will call it aglobal solution.3) If (a1)-(a4), (a6) hold, then the global solution of (2.1) , (2.2) is permanent. Proof.
The proof is illustrated by Fig. 1.1) The existence of a local solution which is positive on [0 , ε ) is justified in the sameway as in [3, 4], using the result of [7, Theorem 4.5, p. 95].This solution is either global or there exists t such that eitherlim inf t → t − x ( t ) = −∞ (2.5)or lim sup t → t − x ( t ) = ∞ (2.6)or either (2.5) or (2.6) is satisfied with y ( t ) instead of x ( t ).The initial value is positive, so as long as x ( t ) > y ( t ) >
0, each component ofthe solution ( x ( t ) , y ( t )) is not less than the solution of the initial value problem for thesystem of ordinary differential equations x ′ ( t ) + r ( t ) x ( t ) = 0 , y ′ ( t ) + r ( t ) y ( t ) = 0 , x ( t ) = x > , y ( t ) = y > , (2.7)and this solution is positive for any t ≥
0. Let us assume that either x ( t ) or y ( t ) becomesnegative and let t be the smallest positive number where either x ( t ) = 0 or y ( t ) = 0.However, the above argument implies x ( t ) ≥ x exp (cid:26)Z t t r ( s ) ds (cid:27) > , y ( t ) ≥ y exp (cid:26)Z t t r ( s ) ds (cid:27) > , which is a contradiction, hence all solutions of (2.1),(2.2) are positive. This also excludesthe possibility that either (2.5) or a similar equality for y ( t ) holds and concludes theproof of Part 1) in the statement of the theorem.2) Assuming (a2 ∗ ), let us prove that (2.6) cannot be satisfied. By the assumptionin (a6), both initial functions are bounded. Fix some ε > ν =max { K + ε, sup s ≤ ϕ ( s ) + ε } , ν = max { f ( K ) + ε, sup s ≤ ψ ( s ) + ε } . Let us verifythat there exist positive bounds M , M for the solutions x and y , respectively, suchthat f ( M ) < M < f − ( M ) and f ( M ) < M < f − ( M ), which means that thepoint ( M , M ) is between the curves f ( x ) (the lower curve) and f − ( x ) (the uppercurve), x > K . tability of cooperative systems with distributed delays f − ( ν ) ≤ ν , denote M = f ( ν ) + ε , where ε < f − ( ν ) − f ( ν ) is a positivenumber, which exists since f − ( x ) − f ( x ) > x > K + ε , and M = ν .If f − ( ν ) > ν but f − ( ν ) ≤ ν , denote M = f − ( ν ) + ε , where ε
0. Let us prove that x ( t ) < M , y ( t ) < M for any t ≥
0. Let usassume the contrary, and let t be the smallest point where either x ( t ) = M or y ( t ) = M . Suppose x ( t ) = M , the case y ( t ) = M is considered similarly.Denote t ∗ = sup { t ∈ [0 , t ] | x ( t ) ≤ f ( M ) } , so x ( t ∗ ) = f ( M ) and x ( t ) > f ( M )for t ∈ ( t ∗ , t ] However, for t ∈ [ t ∗ , t ] we have y ( t ) ≤ M , so f ( y ( t )) ≤ f ( M ) and f ( M ) < x ( t ) < M , thus due to monotonicity of f dxdt = r ( t ) (cid:20)Z th ( t ) f ( y ( s )) d s R ( t, s ) − x ( t ) (cid:21) ≤ r ( t ) [ f ( M ) − f ( M )] = 0 , non-positivity of the derivative of x on [ t ∗ , t ] implies M = x ( t ) ≤ x ( t ∗ ) = f ( M ),which is a contradiction. Thus (2.6) is impossible and there exists a positive globalsolution.3) Next, assume that (a2) holds, which is a particular case of (a2 ∗ ), and provepermanence of equation (2.1) with positive initial conditions.By (a6) we have x = x (0) > y = y (0) >
0, and according to (a3) there is t such that h i ( t ) ≥ t ≥ t , i = 1 ,
2. From positivity of solutions justified in Part 2,there are µ and µ such that x ( t ) ≥ µ > y ( t ) ≥ µ > t ∈ [0 , t ]. Inparticular, we can choose m > m > m < min (cid:8) µ , f − ( µ ) , K (cid:9) , m < min (cid:8) µ , f − ( µ ) , K (cid:9) , (2.8)and also such that the point ( m , m ) is between the curves y = f − ( x ) and y = f ( x ),where 0 < m < K , so f ( m ) > m , f ( m ) > m , (2.9)which is possible since f ( x ) > f − ( x ) for x ∈ [0 , K ] and thus in(0 , min (cid:8) µ , f − ( µ ) , K (cid:9) ). Thus any point ( m , m ) between the curves y = f − ( x )and y = f ( x ), see Fig. 1, satisfies (2.9).Further, let us verify that x ( t ) ≥ m , y ( t ) ≥ m for any t ≥
0. As defined, x ( t ) ≥ m , y ( t ) ≥ m for t ∈ [0 , t ], and also h i ( t ) ≥ t ≥ t , i = 1 ,
2. Thus x ( t ) isgreater than the solution of the ordinary differential equation x ′ ( t ) = r ( t )[ f ( m ) − x ( t )]as long as y ( t ) ≥ m , so x ( t ) is increasing if m < x ( t ) < f ( m ), thus x ( t ) > m unless y ( t ) becomes smaller than m (in fact, even smaller than f − ( m ) < m ). However, y ′ ( t ) > r ( t )[ f ( m ) − y ( t )] tability of cooperative systems with distributed delays x ( t ) ≥ m , thus x ( t ) ≥ m , y ( t ) ≥ m for any t ≥ Corollary 2.4
The results of Theorem 2.3 hold for system (2.3) , (2.2) if instead of (a4)we assume (b4). Corollary 2.5
The results of Theorem 2.3 hold for system (2.4) , (2.2) if assumption(a4) is replaced by (c4). Remark 2.6
Let us note that (a2 ∗ ) guarantees global boundedness but not persistenceof solutions, see Example 2.8 where the solution tends to (0,0) as t → ∞ . The following examples illustrate the fact that when (a2) is not satisfied, thesolution can fail to be either bounded or persistent, even for a non-delay system.
Example 2.7
Let f i ( x ) = x + x , r i ( t ) = 1 , h i ( t ) = t in (2.3) . The system x ′ ( t ) = y ( t ) + y ( t ) − x ( t ) , y ′ ( t ) = x ( t ) + x ( t ) − y ( t ) (2.10) has an unbounded solution ( x ( t ) , y ( t )) = (cid:18) − t , − t (cid:19) on [0 , . The functions f i ( x ) satisfy f ( x ) > x > f − ( x ) , so (a2) does not hold, there is no positive equilibrium. Example 2.8
For f i ( x ) = x , r i ( t ) = 1 , h i ( t ) = t in (2.3) , the system x ′ ( t ) = 12 y ( t ) − x ( t ) , y ′ ( t ) = 12 x ( t ) − y ( t ) (2.11) has a solution ( x ( t ) , y ( t )) = (cid:0) e − t/ , e − t/ (cid:1) on [0 , ∞ ) which tends to zero as t → ∞ andthus is not persistent. The functions f ( x ) = f ( x ) = x satisfy f ( x ) = x < x = f − ( x ) for any x > , thus (a2 ∗ ) is satisfied while (a2) is not.tability of cooperative systems with distributed delays
3. Stability of the Positive Equilibrium
Next, let us proceed to stability. The following result considers the case when f ( x ) > f − ( x ) on (0 , K ), and f ( x ) < f − ( x ) on ( K, ∞ ). This can be interpretedas cooperation for small x and competition for large x . Theorem 3.1 states that in thiscase the equilibrium ( K, f ( K )) attracts all positive solutions. Theorem 3.1
Suppose (a1)-(a6) hold. Then any solution of (2.1) , (2.2) converges tothe unique positive equilibrium ( x ( t ) , y ( t )) → ( K, f ( K )) as t → ∞ . Proof.
The proof is illustrated by Fig. 2.According to Theorem 2.3, there are a, A, b, B such that 0 < a ≤ x ( t ) ≤ A and0 < b ≤ y ( t ) ≤ B for any t ≥
0. We can always assume a < K < A and b < K < B without loss of generality.Consider in addition to f , f a monotone increasing function g : R + → R + whichsatisfies f − ( x ) < g ( x ) < f ( x ) on (0 , K ) and f ( x ) < g ( x ) < f − ( x ) on ( K, ∞ ); inparticular, we can take g ( x ) = αf − ( x ) + (1 − α ) f ( x ), α ∈ (0 , f − ( x ) = 0 if there is no non-negative t such that f ( t ) = x . As we assumed f − (0) = 0,we can always find α ∈ (0 ,
1) such that g (0) ≤ b .Further, let us choose a = min { a, g − ( b ) } , b = min { b, g ( a ) } . The function g ( x )is monotone increasing, and we have either g ( a ) ≥ b or g ( a ) < b . In the former case b = b and a > g − ( b ), so a = g − ( b ) and b = g ( a ). In the latter case b = g ( a ) and a < g − ( b ), so a = a and b = g ( a ).We also have a ≤ a , b ≤ b , so0 < a ≤ x ( t ) ≤ A , < b ≤ y ( t ) ≤ B , t ≥ . (3.1)By (a3), there is t > h i ( t ) ≥ t ≥ t , i = 1 ,
2. Next, we have f ( b ) > b , f ( a ) > a since g ( a ) = b , and the curve g ( x ) is between f − ( x ) and f ( x ), see Fig. 2.Define a = min { g − ( f ( a )) , f ( b ) } , b = min { f ( a ) , g ( f ( b )) } , where b = g ( a ) , f ( y ) ≥ f ( b ) − a ≥ y ∈ [ b , b ] ,f ( x ) ≥ f − ( a ) − b ≥ x ∈ [ a , a ] , (3.2)and the inequalities are strict for any x < a , y < b . Thus (2.1) and (3.2) imply x ′ ( t ) > y ′ ( t ) > x, y ) ∈ [ a , a ] × [ b , b ], and the derivative is positive for any x < a , y < b . Let us prove that there exists t ≥ t such that0 < a ≤ x ( t ) , < b ≤ y ( t ) , t ≥ t . (3.3)Let us choose a ∗ ∈ ( a , a ), b ∗ ∈ ( b , b ) and first prove that there exists t ∗ suchthat x ( t ∗ ) ≥ a ∗ , y ( t ∗ ) ≥ b ∗ . If x ( t ) and y ( t ) satisfy these inequalities, they are alsosatisfied for any t ≥ t due to (3.2), and there is nothing to prove. If either x ( t ) < a ∗ or y ( t ) < b ∗ , or both, then the derivative exceeds a positive value x ′ ( t ) > r ( t )[ f ( b ) − a ∗ ] , y ′ ( t ) > r ( t )[ f − ( a ) − b ∗ ] , tability of cooperative systems with distributed delays x ( t ) < a ∗ , y ( t ) < b ∗ , where the expressions in the brackets are positiveconstants. Due to (a5), there is a point t ∗ such that x ( t ∗ ) ≥ a ∗ , y ( t ∗ ) ≥ b ∗ . Moreover,as (3.1) holds, these inequalities are satisfied for t ≥ t ∗ as well. Let us choose ¯ t suchthat h i ( t ) ≥ t ∗ for t ≥ ¯ t , i = 1 ,
2. Then x ′ ( t ) > r ( t )[ f ( b ∗ ) − a ] , y ′ ( t ) > r ( t )[ f − ( a ∗ ) − b ] , t ≥ ¯ t, as long as x ( t ) < a , y ( t ) < b , and the expressions in the brackets are positive constants.Again, referring to (a5), we obtain that there exists t ∗ ≥ ¯ t such that (3.3) holds.Applying the same procedure to the upper bound, we find t ≥ t ∗ such that0 < a ≤ x ( t ) ≤ A , < b ≤ y ( t ) ≤ B , t ≥ t . (3.4)Continuing the process by induction, we obtain increasing sequences { a n } , { b n } , { t n } and decreasing sequences { A n } , { B n } , where g ( a n ) = b n , g ( A n ) = B n and0 < a n ≤ x ( t ) ≤ A n , < b n ≤ y ( t ) ≤ B n , t ≥ t n . (3.5)Thus all the sequences have limits: lim n →∞ a n = a , lim n →∞ b n = b , and g ( a ) = b ;moreover, all a n < K , b n < K , so a ≤ K , b ≤ K . If a < K then f ( b ) > a and f ( a ) > b , and from continuity there exists ε > f ( x ) > b for x ∈ ( a − ε, a )and f ( y ) > a for y ∈ ( g − ( a − ε ) , b ). As a is a limit, there exists a k ∈ ( a − ε, a ), then a k +1 = min { g − ( f ( a k )) , f ( b k ) } > a , which leads to a contradiction with a > a k for any k . Hence a = K ; similarly, we can prove that A = K and thus any solution of (2.1),(2.2)converges to the unique positive equilibrium: ( x ( t ) , y ( t )) → ( K, f ( K )) as t → ∞ . Corollary 3.2
The results of Theorem 3.1 hold for system (2.4) , (2.2) if assumption(a4) is replaced by (b4).tability of cooperative systems with distributed delays Corollary 3.3
The results of Theorem 3.1 hold for system (2.4) , (2.2) if assumption(a4) is replaced by (c4). Example 3.4
Let us note that condition (a5) is not required for permanence of solutionsbut is crucial for convergence to the unique positive equilibrium. For example, if f i ( x ) = 1 + x , r ( t ) = 2 + sin t , r ( t ) = 2 + cos t , K i ( t, s ) = 1 , h i ( t ) = t − in (2.4) , then all solutions with positive initial values and non-negative initial functions ofthe system x ′ ( t ) = (2 + sin t ) (cid:20)Z tt − (cid:18) y ( s ) (cid:19) ds − x ( t ) (cid:21) ,y ′ ( t ) = (2 + cos t ) (cid:20)Z tt − (cid:20) x ( s ) (cid:19) ds − y ( t ) (cid:21) (3.6) converge to the unique positive equilibrium point (2,2), since all the conditions ofTheorem 3.1 are satisfied.However, system (2.3) with f i ( x ) = 1 + x , r i ( t ) = 2 e t + 0 . , h i ( t ) = t , which is x ′ ( t ) = 2 e t + 0 . (cid:20)(cid:18) y ( t ) (cid:19) − x ( t ) (cid:21) ,y ′ ( t ) = 2 e t + 0 . (cid:20)(cid:18) x ( t ) (cid:19) − y ( t ) (cid:21) , (3.7) has a solution (cid:0) e − t , e − t (cid:1) which tends to (4 , as t → ∞ , not to the uniquepositive equilibrium point (2,2). For system (3.7) with ( x (0) , y (0)) = (5 , , all theconditions of Theorem 3.1 but (a5) are satisfied, since Z ∞ r ( t ) dt = Z ∞ dte t + 0 . < ∞ . In contrast to Theorem 3.1, if f ( x ) < f − ( x ) for any x , all positive solutionsconverge to zero, which can be interpreted as a continuing negative mutual influenceleading to extinction. In the case f ( x ) > f − ( x ) for any x , all positive solutions areunbounded and tend to infinity. The effect is due to mutual positive feedback. Theorem 3.5
Suppose (a1) and (a3)-(a6) hold.1) If f ( x ) < f − ( x ) for any x > then every solution of (2.1) , (2.2) converges tozero as t → ∞ .2) If f ( x ) > f − ( x ) for any x > then every global solution of (2.1) , (2.2) tendsto + ∞ as t → ∞ . Proof.
1) The proof is similar to the proof of Theorem 3.1. First, we notice that thereexist A , B > < x ( t ) ≤ A , < y ( t ) ≤ B , t ≥ t . Next, we define A = f ( B ), B = f ( A ) (see Fig. 3) and prove that for some t > t we have 0 < x ( t ) ≤ A , < y ( t ) ≤ B , t ≥ t . tability of cooperative systems with distributed delays < x ( t ) ≤ A n , < y ( t ) ≤ B n , t ≥ t n where A n = f ( B n − ), B n = f ( A n − ) and both sequences { A n } and { B n } are positive,decreasing and hence have a limit. Let d = lim n →∞ A n , then by construction and continuityof f i we have lim n →∞ B n = f ( d ) and f ( f ( d )) = d , so d = f ( d ) = 0. Thus any solutionof (2.1),(2.2) converges to zero as t → ∞ .The proof of 2) is similar. Corollary 3.6
The results of Theorem 3.5 hold for system (2.3) , (2.2) if assumption(a4) is replaced by (b4). Corollary 3.7
The results of Theorem 3.5 hold for system (2.4) , (2.2) if assumption(a4) is replaced by (c4). Example 3.8
For any h > , consider system (2.4) with f i ( x ) = x + x , r i ( t ) = 1 , h i ( t ) = t − h , K i ( t, s ) = 1 h , with x ′ ( t ) = 1 h Z tt − h (cid:2) y ( s ) + y ( s ) (cid:3) ds − x ( t ) , y ′ ( t ) = 1 h Z tt − h (cid:2) x ( s ) + x ( s ) (cid:3) ds − y ( t ) . (3.8) Every solution with non-negative initial conditions and positive initial values tendsto infinity at the right end of the maximal interval where the solution exists, whichillustrates Part 2 of Theorem 3.5.
Example 3.9
For any h > , consider system (2.4) with f ( x ) = e x − , f ( x ) =12 ln( x + 1) , r i ( t ) = 1 , h i ( t ) = t − h , K i ( t, s ) = 2 h ( s + h − t ) , which is x ′ ( t ) = 2 h Z tt − h ( s + h − t ) (cid:0) e y ( s ) − (cid:1) ds − x ( t ) ,y ′ ( t ) = h R tt − h ( s + h − t ) ln( x ( s ) + 1) ds − y ( t ) . (3.9) tability of cooperative systems with distributed delays Then f − ( x ) = ln( x + 1) > f ( x ) for positive x , by Part 1of Theorem 3.5 every solutionwith non-negative initial conditions and positive initial values tends to zero as t → ∞ . In both Theorems 3.1 and 3.5, all positive solutions had the same asymptotics.Theorem 3.10 considers the case when the limit behaviour depends on the initialconditions. In particular, two cases are considered. In the first case, for small initialconditions, a solution tends to zero, while for large initial conditions, a solution tendsto the unique positive equilibrium. In the second case, for small initial conditions, asolution tends to the unique positive equilibrium, for large initial conditions, a solutiontends to infinity.
Theorem 3.10
Suppose (a1) and (a3)-(a6) hold.1) If f ( x ) < f − ( x ) for any x > , x = K and f ( K ) = f − ( K ) then anysolution of (2.1) , (2.2) with the initial function satisfying ϕ ( t ) ≥ K , ψ ( t ) ≥ f ( K ) tendsto ( K, f ( K )) as t → ∞ , while any solution of (2.1) , (2.2) with the initial functionsatisfying ϕ ( t ) < K , ψ ( t ) < f ( K ) converges to zero as t → ∞ .2) If f ( x ) > f − ( x ) for any x > , x = K and f ( K ) = f − ( K ) then any solutionof (2.1) , (2.2) with the initial function satisfying ϕ ( t ) ≥ K , ψ ( t ) ≥ f ( K ) tends to + ∞ as t → ∞ (or t → c − , where c is a finite right end of the maximal interval ofthe existence of the solution), while any solution of (2.1) , (2.2) with the initial functionsatisfying ϕ ( t ) < K , ψ ( t ) < f ( K ) converges to ( K, f ( K )) as t → ∞ . Proof.
1) The proof of the case ϕ ( t ) ≥ K , ψ ( t ) ≥ f ( K ) completely coincides with theproof of the upper bound in Theorem 3.1, since the lower bound of the solution is K and, as in (a2), f ( x ) < f − ( x ) for x > K . If ϕ ( t ) ∈ (0 , K ), ψ ( t ) ∈ (0 , f ( K )), then werepeat the previous proof for f ( x ) < f − ( x ), where the zero takes the place of K .Similarly, in 2) the part ϕ ( t ) ∈ (0 , K ), ψ ( t ) ∈ (0 , f ( K )) coincides with the proofthat the lower bound tends to K in Theorem 3.1, as f ( x ) > f − ( x ), x ∈ (0 , K ). Forthe proof of the second part we construct a sequence of upper bounds which tends to+ ∞ , as in the proof of Theorem 3.5. Corollary 3.11
The results of Theorem 3.10 hold for system (2.3) , (2.2) if assumption(a4) is replaced by (b4). Corollary 3.12
The results of Theorem 3.10 hold for system (2.4) , (2.2) if assumption(a4) is replaced by (c4). The following result can be interpreted as nonoscillation about the unique positiveequilibrium.
Theorem 3.13
Suppose (a1)-(a4) and (a6) hold. Any solution of (2.1) , (2.2) with theinitial function satisfying ϕ ( t ) ≥ K , ψ ( t ) ≥ f ( K ) satisfies x ( t ) ≥ K , y ( t ) ≥ f ( K ) for any t ≥ , while any solution of (2.1) , (2.2) with ϕ ( t ) ≤ K , ψ ( t ) ≤ f ( K ) satisfies x ( t ) ≤ K , y ( t ) ≤ f ( K ) for any t ≥ .tability of cooperative systems with distributed delays Proof.
Consider the case ϕ ( t ) ≥ K , ψ ( t ) ≥ f ( K ). As long as x ( t ) ≥ K , y ( t ) ≥ f ( K ),we have dxdt ≥ r ( t ) (cid:20)Z th ( t ) f ( f ( K )) d s R ( t, s ) − x ( t ) (cid:21) = r ( t )[ K − x ( t )] ,dydt ≥ r ( t ) (cid:20)Z th ( t ) f ( K ) d s R ( t, s ) − y ( t ) (cid:21) = r ( t )[ f ( K ) − y ( t )] . The first inequality implies x ( t ) ≥ ϕ (0) − K + K exp (cid:26)Z t r ( s ) ds (cid:27) which exceeds K for ϕ (0) > K and is identically equal to K if ϕ (0) = K . The second inequality gives y ( t ) ≥ ψ (0) − f ( K ) + f ( K ) exp (cid:26)Z t r ( s ) ds (cid:27) which is also not less than f ( K ).Again, using monotonicity of f i , the case 0 ≤ ϕ ( t ) ≤ K , 0 ≤ ψ ( t ) ≤ f ( K ), ϕ (0) > ψ (0) > dxdt = r ( t ) G ( x ) (cid:20)Z th ( t ) f ( y ( s )) d s R ( t, s ) − x ( t ) (cid:21) dydt = r ( t ) G ( y ) (cid:20)Z th ( t ) f ( x ( s )) d s R ( t, s ) − y ( t ) (cid:21) (3.10)includes a system of logistic equations with the delay in the production term describedin [1]. We assume that the functions G i satisfy (a7) G i : R + → R + , i = 1 , G i ( x ) > x > Theorem 3.14
Suppose (a1)-(a4),(a6)-(a7) hold. Then any solution of (3.10) , (2.2) ispermanent. Theorem 3.15
Suppose (a1)-(a7) hold. Then any solution of (3.10) , (2.2) converges tothe unique positive equilibrium ( x ( t ) , y ( t )) → ( K, f ( K )) as t → ∞ . Theorem 3.16
Suppose (a1) and (a3)-(a7) hold.1) If f ( x ) < f − ( x ) for any x > then any solution of (3.10) , (2.2) converges tozero as t → ∞ .2) If f ( x ) > f − ( x ) for any x > then any global solution of (3.10) , (2.2) tends toinfinity as t → ∞ . Theorem 3.17
Suppose (a1) and (a3)-(a7) hold.1) If f ( x ) < f − ( x ) for any x > , x = K and f ( K ) = f − ( K ) then anysolution of (3.10) , (2.2) with the initial function satisfying ϕ ( t ) ≥ K , ψ ( t ) ≥ f ( K ) tends to ( K, f ( K )) as t → ∞ , while any solution of (3.10) , (2.2) with the initial functionsatisfying ϕ ( t ) < K , ψ ( t ) < f ( K ) converges to zero as t → ∞ .tability of cooperative systems with distributed delays
2) If f ( x ) > f − ( x ) for any x > , x = K and f ( K ) = f − ( K ) then any solutionof (3.10) , (2.2) with the initial function satisfying ϕ ( t ) ≥ K , ψ ( t ) ≥ f ( K ) tends to + ∞ as t → ∞ (or t → c − , where c is a finite right end of the maximal interval of theexistence of the solution), while any solution of (3.10) , (2.2) with ψ ( t ) < f ( K ) convergesto ( K, f ( K )) as t → ∞ . Theorem 3.18
Suppose (a1)-(a4) and (a6)-(a7) hold. Any solution of (2.1) , (2.2) withthe initial function satisfying ϕ ( t ) ≥ K , ψ ( t ) ≥ f ( K ) satisfies x ( t ) ≥ K , y ( t ) ≥ f ( K ) for any t ≥ , while any solution of (2.1) , (2.2) with ϕ ( t ) ≤ K , ψ ( t ) ≤ f ( K ) satisfies x ( t ) ≤ K , y ( t ) ≤ f ( K ) for any t ≥ . Example 3.19
All solutions of system (2.1) with G i ( x ) = x , r i ( x ) = 1 , R i ( t, s ) = χ ( h i ( t ) , ∞ ) ( s ) , where χ I ( · ) is the characteristic function of set I , x ′ ( t ) = x ( t ) hp y ( h ( t )) + 2 − x ( t ) i , y ′ ( t ) = y ( t ) [ x ( h ( t )) − y ( t )] , with non-negative initial functions and non-trivial initial value converge to the positiveequilibrium (4,4). This is also true for solutions with non-negative initial functions andnon-trivial initial value of system (2.1) with f ( x ) = √ x + 2 , f ( x ) = x , r i ( t ) = 1 , h i ( t ) = t − h , R i ( t, s ) = h ( s − t + h ) if s ∈ [ t − h, t ] and zero elsewhere, which is x ′ ( t ) = x ( t ) h Z tt − h ( p y ( s ) + 2) ds − x ( t ) , y ′ ( t ) = y ( t ) h Z tt − h x ( s ) ds − y ( t ) .
4. Applications and Discussion
As an example, consider the following models of type (1.3) dxdt = c Z th ( t ) K ( t, s ) tanh( y ( s )) ds − µ x ( t ) dydt = c Z th ( t ) K ( t, s ) tanh( x ( s )) ds − µ y ( t ) (4.1)and dxdt = c tanh( y ( h ( t )) − µ x ( t ) , dydt = c tanh( x ( h ( t )) − µ y ( t ) , (4.2)where c , c , µ , µ are positive constants, Z th ( t ) K ( t, s ) ds = Z th ( t ) K ( t, s ) ds ≡ K i ( t, s ) ≥ t ≥ s > h i are Lebesgue measurable functions satisfying h i ( t ) ≥ t →∞ h i ( t ) = ∞ , i = 1 ,
2. A more general version of (4.1) but withconstant delays was studied in [8].Since for u > f i ( u ) = c i tanh( u ) /µ i satisfy f ′ i ( u ) > f ′′ i ( u ) < f ′ (0) > /f ′ (0), or c /µ > µ /c , otherwise f ( x ) < f − ( x ) for x > tability of cooperative systems with distributed delays c c > µ µ , then all solutions of (4.1),(2.2) and (4.2),(2.2) with non-negativeinitial functions and positive initial values converge to the unique positive equilibrium( x ∗ , y ∗ ), where x ∗ is a solution of the equation c µ tanh (cid:18) c µ tanh( x ∗ ) (cid:19) = x ∗ ,y ∗ = c tanh( x ∗ ) /µ . If c c ≤ µ µ , then all solutions of (4.1),(2.2) and (4.2),(2.2)converge to zero.Next, for the Lotka-Volterra-type cooperative system dxdt = r ( t ) x ( t ) (cid:20) A − a x ( t ) + b Z th ( t ) y ( s ) d s R ( t, s ) (cid:21) dydt = r ( t ) y ( t ) (cid:20) A − a y ( t ) + b Z th ( t ) x ( s ) d s R ( t, s ) (cid:21) (4.3)Theorems 3.1 and 3.5 imply the following result. Theorem 4.1
Suppose (a3)-(a6) hold, A i ≥ , a i > and b i > , i = 1 , .If A + A > and a a < b b then there exists a unique positive equilibrium (( b A − a A ) / ( b b − a a ) , ( b A − a A ) / ( b b − a a )) of (4.3) , and all solutions of (4.3) , (2.2) converge to this equilibrium. If A = A = 0 and a a < b b all solutions of (4.3) , (2.2) converge to (0,0). If A + A > and a a ≥ b b then both components ofthe solution of (4.3) , (2.2) tend to + ∞ as t → ∞ . Next, let us consider the generalization of the cooperative model [14, p.192] to thecase of distributed delays and time-variable growth rates dxdt = r ( t ) x ( t ) (cid:20)Z th ( t ) K + α y ( s )1 + y ( s ) d s R ( t, s ) − x ( t ) (cid:21) dydt = r ( t ) y ( t ) (cid:20)Z th ( t ) K + α x ( s )1 + x ( s ) d s R ( t, s ) − y ( t ) (cid:21) (4.4)The following result generalizes [14, Theorem 3.3.4, p. 193]. Theorem 4.2
Suppose (a3)-(a6) hold and α i > K i , i = 1 , .Then there exists a unique positive equilibrium of (4.4) , and all solutions of (4.4) , (2.2) converge to this equilibrium. A natural generalization of the results of the present paper would be to n -dimensional cooperative systems, as well as models with general nonlinear non-delaymortality dxdt = r ( t ) (cid:20)Z th ( t ) f ( y ( s )) d s R ( t, s ) − g ( x ( t )) (cid:21) dydt = r ( t ) (cid:20)Z th ( t ) f ( x ( s )) d s R ( t, s ) − g ( y ( t )) (cid:21) tability of cooperative systems with distributed delays f i , such models have the sameproperties as equations with linear mortality functions g i ( x ) = b i x [3]. So farwe considered the case of the unique coexistence equilibrium; however, it wouldbe interesting to study multiple coexistence equilibria. For a single equation thisinvestigation was implemented in [4]. Acknowledgments
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