STABILITY OF A POSITIVE POINT OF EQUILIBRIUM OF ONE NONLINEAR SYSTEM WITH AFTEREFFECT AND STOCHASTIC PERTURBATIONS

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1 Dynamic Systems and Applications STABILITY OF A POSITIVE POINT OF EQUILIBRIUM OF ONE NONLINEAR SYSTEM WITH AFTEREFFECT AND STOCHASTIC PERTURBATIONS LEONID SHAIKHET Department of Higher Mathematics, Donetsk State University of Management, Chelyuskintsev 163-a, Donetsk 8315, Ukraine ABSTRACT. The aim of the paper is to show one useful way for stability investigation of the positive point of equilibrium of some nonlinear system with aftereffect and stochastic perturbations. Obtained results are applied for stability investigation of some mathematical predator-prey models. AMS MOS Subject Classification. 34D2. 1. INTRODUCTION The study of realistic mathematical models in ecology, especially the study of relations between species and their environment has become a very popular topic that interested both mathematicians and biologists. Investigations on various population models reflect their use in helping to understand the dynamic processes involved in such areas as predator-prey and competition, renewable resource management, evolution of pesticide resistant strains, ecological control of pests, multispecies societies, plant-herbivore systems, and so on. Well known Lotka-Volterra predator-prey mathematical model after its appearance got specially wide development in many different directions, in particular, for systems with delays and stochastic perturbations [1, 3, 1, 11, 12, 16, 17, 18, 27, 28, 33, 34, 35, 36]. In this paper some general nonlinear mathematical model is considered that is destined to unify different known models, in particular, the models type of predatorprey. The following method for stability investigation of the positive point of equilibrium is proposed. The system under consideration is exposed to stochastic perturbations and is linearized in the neighborhood of the positive point of equilibrium. Asymptotic mean square stability conditions are obtained for the constructed linear system. In the case if the order of nonlinearity more than 1 these conditions are sufficient ones [2, 3, 31, 32] for stability in probability of the initial nonlinear system by stochastic perturbations. Received December 27, $15. c Dynamic Publishers, Inc.

2 236 L. SHAIKHET This method can be successfully used for stability investigation of many other types of known biological systems: stage-structure predator-prey models [9, 15, 37], SIR epidemic models [2, 6], chemostat models [4, 5] and other [7, 8, 14, 25, 26] SYSTEM UNDER CONSIDERATION INTRODUCTION Consider the system of two nonlinear differential equations ẋ 1 t = x 1 t a F x 1t, x 2t F 1 x 1t, x 2t, ẋ 2 t = x 2 t b + G x 1t, x 2t + G 1 x 1t, x 2t, x i s = φ i s, s, i = 1, 2. Here x i t, i = 1, 2, is a value of process x i in the point of time t and x it = x i t + s, s, is a trajectory of the process x i to the point of t. 2.2 Put, for example, F x 1t, x 2t = f x 1 t sdk s, F 1 x 1t, x 2t = 2 i=1 f i x i t sdk i s, G x 1t, x 2t = G 1 x 1t, x 2t = 2 i=1 g x 1 t sdr s, g i x i t sdr i s, where K i s and R i s, i =, 1, 2, are nondecreasing functions, such that 2.3 K i = dk i s <, R i = dr i s <, ˆK i = sdk i s <, ˆRi = sdr i s <, and all integrals are understanding in Stieltjes sense. In this case system 2.1 takes the form 2.4 ẋ 1 t = x 1 t a f x 1 t sdk s 2 i=1 ẋ 2 t = x 2 t b + g x 1 t sdr s + 2 i=1 f i x i t sdk i s, g i x i t sdr i s. Systems type of 2.1 are investigated in some biological problems. Put here, for example, 2.5 δs is Dirac s function. f x = f 1 x = f 2 x = g 1 x = g 2 x = x, g x =, dk 1 s = δsds, dr s =, If a and b are positive constants, x 1 t and x 2 t are respectively the densities of prey and predator populations then 2.4 is transformed to the mathematical predator-prey model [32] 2.6 Putting in 2.6 ẋ 1 t = x 1 t a x 1 t sdk s x 2 t sdk 2 s, ẋ 2 t = bx 2 t + x 1 t sdr 1 s x 2 t sdr 2 s. 2.7 dk s = a 1 δsds, dk 2 s = a 2 δsds, dr i s = b i δs h i ds, i = 1, 2,

3 STABILITY OF A POSITIVE POINT OF EQUILIBRIUM 237 we obtain the known predator-prey mathematical model with delays 2.8 ẋ 1 t = x 1 ta a 1 x 1 t a 2 x 2 t, ẋ 2 t = bx 2 t + b 1 b 2 x 1 t h 1 x 2 t h 2. If here h 1 = h 2 = we have the classical Lotka-Volterra model 2.9 ẋ 1 t = x 1 ta a 1 x 1 t a 2 x 2 t, ẋ 2 t = x 2 t b + b 1 b 2 x 1 t. Many authors [1, 3, 9, 11, 12, 34, 35] consider ratio-dependent predator-prey models with delays type of 2.1 ẋ 1 t = x 1 t a x 1 t sdk s ẋ 2 t = bx 2 t + x m 1 x 2t x m 1 +b 2x mdr 1s. 2 Here it is supposed that m and k are positive constants System 2.1 follows from 2.1 if x k 1 x 2t x k 1 +a 2x k 2 dk 1s, F x 1t, x 2t = x 1 t sdk s, F 1 x 1t, x 2t = fx 1 t s, x 2 t sx 2 tdk 1 s, G 1 x 1t, x 2t = gx 1 t s, x 2 t sx 2 tdr 1 s, fx 1, x 2 = xk 1 x k 1 +a 2x k 2 Putting in 2.1, for example, 2.12 we obtain the system 2.13, gx 1, x 2 = xm 1 x m 1 +b 2x m 2 dk s = a δsds, dk 1 s = a 1 δsds, k = 1, dr 1 s = b 1 δs hds, m = 1, ẋ 1 t = x 1 t ẋ 2 t = x 2 t that was considered in [3, 9]. a a x 1 t b + a 1x 2 t x 1 t+a 2 x 2 t b 1 x 1 x 1 +b 2 x 2 3. POSITIVE POINT OF EQUILIBRIUM, STOCHASTIC PERTURBATIONS, CENTERING AND LINEARIZATION 3.1. Let in system 2.1 F i = F i φ, ψ and G i = G i φ, ψ, i =, 1, be functionals defined on H H, where H be a set of functions φ = φs, s, with the norm φ = sup s φs, the functionals F i and G i be nonnegative ones for nonnegative functions φ and ψ. Let us suppose also that system 2.1 has a positive point x 1, x 2 of equilibrium. This point is obtained from the conditions ẋ 1 t, ẋ 2 t and is defined by the system of algebraic equations,., 3.1 x 1a F x 1, x 2 = F 1 x 1, x 2, x 2 b + G x 1, x 2 = G 1x 1, x 2.

4 238 L. SHAIKHET Note that system 3.1 has a positive solution only by the condition 3.2 a > F x 1, x 2. For example, for system 2.4 a positive point of equilibrium is defined by the equations 3.3 x 1a K f x 1 = K 1 K 2 f 1 x 1f 2 x 2, x 2b + R g x 1 = R 1 R 2 g 1 x 1g 2 x 2, if a > K f x 1. In particular, in the case f x = f 1 x = f 2 x = g x = g 1 x = g 2 x = x system 3.3 has the positive solution 3.4 x 1 = by the condition b R 1 R 2 R, x 2 = a K x 1 K 1 K 2 x 1 = A K, x 2 = A BK, a > K b R 1 R 2 R >. = a K R 1 R 2 R 1 b K 1 K 2, For system 2.1 the positive point of equilibrium is defined as follows K 1 b2 b A = a >, B = B + a 2 B1 k R 1 b In particular, for system 2.13 it is 1 m >. 3.5 x 1 = A a, x 2 = A Ba, A = a a 1 B + a 2 >, B = bb 2 b 1 b > As it was proposed in [2, 32] and used later in [1, 8] let us assume that system 2.1 is exposed to stochastic perturbations, which are of white noise type, are directly proportional to the deviations of x 1 t and x 2 t from the values of x 1, x 2 and influence on ẋ 1t, ẋ 2 t respectively. In this way system 2.1 is transformed to the form 3.6 ẋ 1 t = x 1 t a F x 1t, x 2t F 1 x 1t, x 2t + σ 1 x 1 t x 1 ẇ 1t, ẋ 2 t = x 2 t b + G x 1t, x 2t + G 1 x 1t, x 2t + σ 2 x 2 t x 2 ẇ 2t. Here σ 1, σ 2 are constants, w 1, w 2 are independent of each other Wiener processes [13] Centering system 3.6 on the positive point of equilibrium via new variables y 1 = x 1 x 1, y 2 = x 2 x 2, we obtain ẏ 1 t = y 1 t + x 1 a F y 1t + x 1, y 2t + x 2 F 1 y 1t + x 1, y 2t + x 2 + σ 1y 1 tẇ 1 t, 3.7 ẏ 2 t = y 2 t + x 2 b + G y 1t + x 1, y 2t + x 2 +G 1 y 1t + x 1, y 2t + x 2 + σ 2y 2 tẇ 2 t. It is clear that stability of system 3.6 equilibrium x 1, x 2 is equivalent to stability of the trivial solution of system 3.7.

5 STABILITY OF A POSITIVE POINT OF EQUILIBRIUM Along with system 3.6 we will consider the linear part of this system. Let us suppose that the functions f i x, g i x, i =, 1, 2, in system 2.4 are differentiable ones. Using for all these functions the representation fz + x = f + f 1 z + oz, f = fx, f 1 = df dx x, and neglecting by oz, via 2.2, 3.7 we obtain the linear part process z 1 t, z 2 t of system 2.4 after adding stochastic perturbations and centering 3.8 where 3.9 ż 1 t = a K f z 1 t z 1 t sdks K 1 f 1 f 21 z 2 t sdk 2 s + σ 1 z 1 tẇ 1 t, ż 2 t = b + R g z 2 t + z 1 t sdrs +R 1 g 1 g 21 z 2 t sdr 2 s + σ 2 z 2 tẇ 2 t, dks = K 2 f 2 f 11 dk 1 s + f 1 x 1 dk s, drs = R 2 g 2 g 11 dr 1 s g 1 x 2 dr s. Below we will say about system 3.8 also as about the linear part appropriate to system 2.4 or for brevity as about the linear part of system 2.4. In particular, by conditions 2.5, 3.3 from 3.8, 3.9 we obtain the linear part of system 2.6 ż 1 t = x 1 z 1 t sdk s σ 1 z 1 tẇ 1 t, ż 2 t = bz 2 t + R 2 x 2 +R 1 x 1 z 1 t sdr 1 s z 2 t sdk 2 s z 2 t sdr 2 s + σ 2 z 2 tẇ 2 t. Via 2.7 from 3.1 we have the linear part of system ż 1 t = x 1a 1 z 1 t + a 2 z 2 t + σ 1 z 1 tẇ 1 t, ż 2 t = bz 2 t + b 1 b 2 x 2z 1 t h 1 + x 1z 2 t h 2 + σ 2 z 2 tẇ 2 t. 4. AUXILIARY STATEMENTS We will use two definitions of stability. Definition 4.1. The trivial solution of system 3.7 is called stable in probability if for any ɛ 1 > and ɛ 2 > there exists δ > such that the solution yt = yt, φ, where y = y 1, y 2, φ = φ 1, φ 2, satisfies P{sup t yt, φ > ɛ 1 } < ɛ 2 for any initial function φ H satisfying P{ φ δ} = 1.

6 24 L. SHAIKHET Definition 4.2. The trivial solution of system 3.8 is called mean square stable if for any ɛ > there exists δ > such that the solution zt = zt, φ, where z = z 1, z 2, φ = φ 1, φ 2, satisfies E zt, φ 2 < ɛ for any initial function φ H such that sup s E φs 2 < δ. If besides lim t E zt, φ 2 = for any initial function φ H then the trivial solution of equation 3.8 is called asymptotically mean square stable. As it is shown in [28, 29] if the order of nonlinearity of the system under consideration is more than 1 then a sufficient condition for asymptotic mean square stability of the linear part of the initial nonlinear system is also a sufficient condition for stability in probability of the initial system. So, in this paper we will obtain sufficient conditions for asymptotic mean square stability of the linear part of considered nonlinear systems. We will use also the following auxiliary statements. Theorem 4.3. [19] Let there exists a functional V t, ϕ, t, ϕ H, such that c 1 E ϕ 2 EV t, ϕ c 2 sup E ϕs 2, s ELV t, ϕ c 3 E ϕ 2, where L is the generator [13] of equation 3.7, c i >, i = 1, 2, 3. Then the trivial solution of system 3.7 is asymptotically mean square stable. Consider system of stochastic differential equations without delays 4.1 u 1 t = α 1 u 1 t + α 2 u 2 t + σ 1 u 1 tẇ 1 t, u 2 t = β 1 u 1 t + β 2 u 2 t + σ 2 u 2 tẇ 2 t, and put 4.2 ɛ i = 1 2 σ2 i, i = 1, 2. Lemma 4.4. Let there exist numbers µ and γ satisfying the conditions 4.3 γ > µ 2, α 1 + µβ 1 + ɛ 1 <, µα 2 + γβ 2 + ɛ 2 <, 4.4 4α 1 + µβ 1 + ɛ 1 µα 2 + γβ 2 + ɛ 2 > µα 1 + β 2 + α 2 + γβ 1 2. Then the trivial solution of system 4.1 is asymptotically mean square stable. Proof. Let L be the generator [13] of system 4.1. Using the function 4.5 vt = u 2 1t + 2µu 1 tu 2 t + γu 2 2t and 4.2 for system 4.1 we have L vt = 2u 1 t + µu 2 tα 1 u 1 t + α 2 u 2 t +2µu 1 t + γu 2 tβ 1 u 1 t + β 2 u 2 t + σ 2 1 u2 1 t + γσ2 2 u2 2 t

7 STABILITY OF A POSITIVE POINT OF EQUILIBRIUM 241 = 2α 1 + µβ 1 + ɛ 1 u 2 1 t + 2µα 2 + γβ 2 + ɛ 2 u 2 2 t µα 1 + β 2 + α 2 + γβ 1 u 1 tu 2 t. By conditions 4.3, 4.4 from 4.5, 4.6 it follows that function 4.5 is a positive definite one and L vt is a negative definite one, i.e. function 4.5 satisfies Theorem 4.3. Thus, the trivial solution of system 4.1 is asymptotically mean square stable. Lemma is proven. Corollary 4.5. Suppose that the parameters of system 4.1 satisfy the conditions β 1 >, α 1 + β 2 β 2 + ɛ 2 > β 1 α 2, 4.7 α 2 β 2 + ɛ 2 α 1 + β 2 β 2 + ɛ 2 β 1 α 2 > α 1 + ɛ 1 β 1, α 1 + β 2 2 > 4β 1 α 2, and the intervals 4.8 α 2 β 2 + ɛ 2, α 1 + ɛ 1, α 1 + β 2 β 2 + ɛ 2 β 1 α 2 β α 1 + β 2 2 4β 1 α 2 α 1 + β 2 2β 1, α1 + β 2 2 4β 1 α 2 α 1 + β 2 2β 1 have common points. Then the trivial solution of system 4.1 is asymptotically mean square stable. Proof. Choose γ from the condition µα 1 + β 2 + α 2 + γβ 1 =. To satisfy the conditions 4.3, 4.4 µ has belong to intervals 4.8, 4.9. Lemma 4.6. For positive P 2, x, y and nonnegative P 1, Q 1, Q 2, such that P 2 > Q 1 x + Q 2 y the following inequality holds P 1 + Q 1 x 1 + Q 2 y 1 P 2 Q 1 x Q 2 y 2 Q1 + Q P 1 P 2 + Q 1 + Q 2. P 2 Proof. It is enough to show that the function reaches its minimum in the point x = y = fx, y = P 1 + Q 1 x 1 + Q 2 y 1 P 2 Q 1 x Q 2 y P 2 Q1 + Q P 1 P 2 + Q 1 + Q 2. Lemma is proven.

8 242 L. SHAIKHET 5. STABILITY OF EQUILIBRIUM OF SYSTEM 2.4 WITH STOCHASTIC PERTURBATIONS Consider system 3.8 as the linear part of system 2.4 with stochastic perturbations. Following the general method of Lyapunov functionals construction GMLFC [2, 21, 22, 23, 24, 29] rewrite system 3.8 in the form 5.1 where 5.2 Z 1 t = z 1 t Z 2 t = z 2 t + and via 3.3, 3.9 Ż 1 t = α 1 z 1 t + α 2 z 2 t + σ 1 z 1 tẇ 1 t, Ż 2 t = β 1 z 1 t + β 2 z 2 t + σ 2 z 2 tẇ 2 t, z 1θdθdKs K 1 f 1 f 21 z 1θdθdRs + R 1 g 1 g 21 z 2θdθdK 2 s, z 2θdθdR 2 s, 5.3 f α 1 = a K K f = K 1 K 2 f 1 2 f x 11 K f 1 x 1, 1 α 2 = K 1 K 2 f 1 f 21, β 1 = R = R 1 R 2 g 2 g 11 R g 1 x 2, β 2 = R 1 R 2 g 1 g 21 b R g = R 1 R 2 g 1 g2 x 2 g 21. We will suppose that the corresponding auxiliary system without delays 4.1 with α i, β i, i = 1, 2, defined by 5.3, is asymptotically mean square stable. Following GMLFC, we have construct Lyapunov functional V for system 5.1 in the form V = V 1 + V 2, where in corresponding with V 1 t = Z1t 2 + 2µZ 1 tz 2 t + γz2t 2 and V 2 has be chosen by some standard way after estimation of LV 1, where L is the generator of system 5.1. Via 5.1 LV 1 t = 2Z 1 t + µz 2 tα 1 z 1 t + α 2 z 2 t µZ 1 t + γz 2 tβ 1 z 1 t + β 2 z 2 t + σ 2 1z 2 1t + γσ 2 2z 2 2t = 2α 1 + µβ 1 Z 1 tz 1 t + 2µα 1 + γβ 1 Z 2 tz 1 t +2α 2 + µβ 2 Z 1 tz 2 t + 2µα 2 + γβ 2 Z 2 tz 2 t + σ 2 1 z2 1 t + γσ2 2 z2 2 t. Substituting 5.2 into 5.5 and using 4.2, we have LV 1 = 2α 1 + µβ 1 + ɛ 1 z 2 1 t + 2µα 2 + γβ 2 + ɛ 2 z 2 2 t +2µα 1 + β 2 + α 2 + γβ 1 z 1 tz 2 t 2α 1 + µβ 1 2α 1 + µβ 1 K 1 f 1 f 21 z 1 tz 1 θdθdks z 1 tz 2 θdθdk 2 s +2µα 1 + γβ 1 z 1 tz 1 θdθdrs

9 From here, using 2.3, STABILITY OF A POSITIVE POINT OF EQUILIBRIUM µα 1 + γβ 1 R 1 g 1 g 21 2α 2 + µβ 2 2α 2 + µβ 2 K 1 f 1 f 21 z 2 tz 1 θdθdks z 1 tz 2 θdθdr 2 s z 2 tz 2 θdθdk 2 s +2µα 2 + γβ 2 z 2 tz 1 θdθdrs +2µα 2 + γβ 2 R 1 g 1 g 21 z 2 tz 2 θdθdr 2 s. ˆK = s dks, ˆR = and some positive numbers γ i, i = 1,..., 4, we obtain LV 1 2α 1 + µβ 1 + ɛ 1 z 2 1 t + 2µα 2 + γβ 2 + ɛ 2 z 2 2 t s drs, or +2µα 1 + β 2 + α 2 + γβ 1 z 1 tz 2 t + ˆK α 1 + µβ 1 z 2 1 t + α 1 + µβ 1 +γ 1 K 1 ˆK2 f 1 f 21 α 1 + µβ 1 z 2 1 t +γ 1 1 K 1 f 1 f 21 α 1 + µβ 1 + ˆR µα 1 + γβ 1 z 2 1 t + µα 1 + γβ 1 +γ 2 R 1 ˆR2 g 1 g 21 µα 1 + γβ 1 z 2 1 t +γ 1 2 R 1g 1 g 21 µα 1 + γβ 1 +γ 1 3 ˆK α 2 + µβ 2 z 2 2t + γ 3 α 2 + µβ 2 z 2 2θdθdK 2 s z 2 1 θdθ dks z 2 2 θdθdr 2s z 2 1 θdθ drs +K 1 ˆK2 f 1 f 21 α 2 + µβ 2 z 2 2 t + K 1f 1 f 21 α 2 + µβ 2 +γ 1 4 ˆR µα 2 + γβ 2 z 2 2 t + γ 4 µα 2 + γβ 2 +R 1 ˆR2 g 1 g 21 µα 2 + γβ 2 z 2 2t + R 1 g 1 g 21 µα 2 + γβ 2 z 2 1θdθ dks z 2 1 θdθ drs z 2 2 θdθdk 2s z 2 2θdθdR 2 s LV 1 [2α 1 + µβ 1 + ɛ 1 + ˆK α 1 + µβ 1 + γ 1 K 1 ˆK2 f 1 f 21 α 1 + µβ 1 + ˆR µα 1 + γβ 1 + γ 2 R 1 ˆR2 g 1 g 21 µα 1 + γβ 1 ]z 2 1 t

10 244 L. SHAIKHET 5.6 where +[2µα 2 + γβ 2 + ɛ 2 + γ 1 3 ˆK α 2 + µβ 2 + K 1 ˆK2 f 1 f 21 α 2 + µβ 2 +γ 1 4 ˆR µα 2 + γβ 2 + R 1 ˆR2 g 1 g 21 µα 2 + γβ 2 ]z2 2 t 2 +2[µα 1 + β 2 + α 2 + γβ 1 ]z 1 tz 2 t + zi 2 θdθdf is, i=1 Then df 1 s = α 1 + µβ 1 + γ 3 α 2 + µβ 2 dks + µα 1 + γβ 1 + γ 4 µα 2 + γβ 2 drs, df 2 s = K 1 f 1 f 21 γ 1 1 α 1 + µβ 1 + α 2 + µβ 2 dk 2 s +R 1 g 1 g 21 γ 1 2 µα 1 + γβ 1 + µα 2 + γβ 2 dr 2 s. Following GMLFC, the additional functional V 2 we have to choose in the form 2 V 2 t = θ t + szi 2 θdθdf is. i=1 5.7 LV 2 t = ˆF 1 z 2 1 t + ˆF 2 z 2 2 t 2 where i=1 ˆF 1 = α 1 + µβ 1 + γ 3 α 2 + µβ 2 ˆK + µα 1 + γβ 1 + γ 4 µα 2 + γβ 2 ˆR, z 2 i θdθdf is, ˆF 2 = K 1 f 1 f 21 γ 1 1 α 1 + µβ 1 + α 2 + µβ 2 ˆK 2 +R 1 g 1 g 21 γ 1 2 µα 1 + γβ 1 + µα 2 + γβ 2 ˆR 2. Via 5.6, 5.7 the functional V = V 1 + V 2 satisfies the condition LV t z tp zt, where 5.8 zt = z 1 t z 2 t p 11 = 2α 1 + µβ 1 + ɛ 1, P = p 11 p 12 p 12 p 22, ˆK + γ 1 K 1 ˆK2 f 1 f 21 α 1 + µβ 1 +2 ˆR + γ 2 R 1 ˆR2 g 1 g 21 µα 1 + γβ 1 +γ 3 ˆK α 2 + µβ 2 + γ 4 ˆR µα 2 + γβ 2, p 22 = 2µα 2 + γβ 2 + ɛ 2 +γ 1 1 K 1 ˆK 2 f 1 f 21 α 1 + µβ 1 + γ 1 2 R 1 ˆR 2 g 1 g 21 µα 1 + γβ 1 +γ 1 3 ˆK + 2K 1 ˆK2 f 1 f 21 α 2 + µβ 2 +γ 1 4 ˆR + 2R 1 ˆR2 g 1 g 21 µα 2 + γβ 2,

11 STABILITY OF A POSITIVE POINT OF EQUILIBRIUM 245 p 12 = µα 1 + β 2 + α 2 + γβ 1. Corollary 5.1. If there exist numbers µ, γ > µ 2, γ i >, i = 1, 2, 3, 4, such that the matrix P, defined by 5.8, 5.9, is negative definite one, i.e. p 11 <, p 22 <, p 11 p 22 > p 2 12, then the trivial solution of system 3.8 is asymptotically mean square stable and the positive point of equilibrium of system 2.4 with stochastic perturbations is stable in probability. To simplify the obtained stability condition consider some particular cases of system 3.8. Via 2.5, 3.9, 5.3 for system 3.1 that is the linear part of system 2.6 with stochastic perturbations we have: 5.1 α 1 = K x 1 <, α 2 = K 2 x 1 <, β 1 = R 1 R 2 x 2 >, β 2 =, dks = K 2 x 2 δsds + x 1 dk s, drs = R 2 x 2 dr 1s, K 1 = 1, ˆK = ˆK x 1, ˆR = ˆR 1 R 2 x 2. Choosing γ from the condition p 12 =, we obtain 5.11 γ = µα 1 + α 2 β 1 = µ α 1 + α 2 β 1. From the condition p 11 < it follows α 1 + µβ 1 <. So, using 5.9, 5.1, 5.11, we have p 11 = 2 α 1 + µβ 1 + ɛ ˆK + γ 1 ˆK2 x 1 α 1 µβ 1 +2 ˆR 1 R 2 x 2 + γ 2 R 1 ˆR2 x 1 α 2 + γ 3 ˆK x 1 α 2 + µγ 4 ˆR1 R 2 x 2 α 2, µ α 1 + α 2 p 22 = 2 µ α 2 + ɛ 2 + γ1 1 ˆK 2 x β 1 α 1 µβ 1 + γ2 1 R 1 ˆR2 x 1 α 2 1 +γ3 1 ˆK + 2 ˆK 2 x 1 α 2 + µγ4 1 ˆR 1 R 2 x 2 + 2R ˆR 1 2 x 1 α 2. From the conditions p 11 <, p 22 < it follows 5.12 where 5.13 A 1 + B 1 γ1 1 + B 2 γ2 1 + B 3 γ3 1 A 4 + B γ1 1 B 4 γ4 1 < µ < A 2 B 1 γ 1 B 2 γ 2 B 3 γ 3 A 3 B γ 1 + B 4 γ 4, A 1 = 2 α 2 ˆK 2 x 1 + ɛ 2β1 1, A 2 = 2[ α 1 1 ˆK x 1 ˆR 1 R 2 α 2 x 2 ɛ 1], A 3 = 2β 1 1 ˆK x 1, A 4 = 2[ α 2 1 R 1 ˆR2 x 1 ɛ 2 α 1 β1 1 ], B = ˆK 2 β 1 x 1, B 1 = ˆK 2 α 1 x 1, B 2 = R 1 ˆR2 α 2 x 1, B 3 = ˆK α 2 x 1, B 4 = ˆR 1 R 2 α 2 x 2.

12 246 L. SHAIKHET Via 5.12 we can write A1 + B 1 γ1 1 + B 2 γ2 1 + B 3 γ 1 3 A3 B γ 1 + B 4 γ A 2 B 1 γ 1 B 2 γ 2 B 3 γ 3 A 4 + B γ1 1 B 4 γ4 1 < 1 and minimize the left part of inequality 5.14 with respect to γ 1, γ 2, γ 3, γ 4 conditions by 5.15 A 2 > B 1 γ 1 + B 2 γ 2 + B 3 γ 3, A 3 + B 4 γ 4 > B γ 1, A 4 + B γ 1 1 > B 4 γ 1 4. Corollary 5.2. If minimum of the left part of inequality 5.14 with respect to γ 1, γ 2, γ 3, γ 4 by conditions 5.15 less than 1 then the trivial solution of system 3.1 is asymptotically mean square stable and the positive equilibrium of system 2.6 with stochastic perturbations is stable in probability. Let us show that in some cases minimum of the left part of inequality 5.14 can be easy obtained. Suppose, for example, that ˆK 2 =. From 5.13 it follows that B = B 1 = and A 1 = 2 α 2 ɛ 2 β1 1. Inequality 5.14 takes the form A1 + B 2 γ2 1 + B 3 γ3 1 A3 + B 4 γ 4 A 2 B 2 γ 2 B 3 γ 3 A 4 B 4 γ4 1 < 1 Note µ from inequalities 5.12 satisfies the condition γ > µ 2 that via 5.1, 5.11 is equivalent to β 1 µ 2 µ α 1 α 2 < or α 1 α β 1 α 2 2β 1 < µ < µ = α 1 + Really, from 5.12, 5.15 via B = B 1 = it follows < µ < A 2 A 3 < α 1 β 1 < µ. α β 1 α 2 2β 1. Using Lemma 4.2 firstly for P 1 = A 1, P 2 = A 2, Q 1 = B 2, Q 2 = B 3, x = γ 2, y = γ 3 and secondly for P 1 = A 3, P 2 = A 4, Q 1 = B 4, Q 2 =, x = γ 1 4, we obtain the following Corollary 5.3. Let A 1, A 2, A 3, A 4, B 2, B 3, B 4 are defined by If ˆK2 =, A 2 >, A 3, A 4 > and B2 + B A 1 A 2 + B 2 + B 3 B 24 + A 3 A 4 + B 4 < A 2 A 4 then the trivial solution of equation 3.1 is asymptotically mean square stable and the positive point of equilibrium of system 2.6 is stable in probability. Example 5.4. Suppose that the parameters of system 2.8 that is a particular case of system 2.6 satisfy the condition 5.16 A = a a 1b b 1 b 2 >.

13 STABILITY OF A POSITIVE POINT OF EQUILIBRIUM 247 From 3.4 via 2.3, 2.5, 2.7 it follows that the positive point of equilibrium of system 2.8 is x 1 = b, x 2 = A. b 1 b 2 a 2 The linear part of system 2.8 with stochastic perturbations is defined by Via 2.7, 5.1, 5.13 we obtain and 5.17 A 1 = ba2 2σ2 2, A 2 = 2b A 4 = a 2b b 1 b 2 Ab 2 1 b2 2 Via Corollary 5.3 if a1 21 bh 2 a 1σ 2 2 Ab 1 b 2 h 1 < 1 A Ah 1 b 1 b 2, B 2 = a 2b 2 σ 2 1, A 3 = 2Ab 1b 2 a 2, b 1 b 2 h 2, B 4 = Abh 1. a1 σ2 1, h 2 < 1 1 a 1σ2 2 b 1 b 2 2b b 2Ab 1 b 2 B 22 + A 1 A 2 + B 2 B 24 + A 3 A 4 + B 4 < A 2 A 4 then the trivial solution of equation 3.11 is asymptotically mean square stable and the positive point of equilibrium of system 2.8 is stable in probability. Note that in the case h 1 = h 2 = the obtained stability condition follows from Corollary 5.1 via α 1 = a 1 x 1, α 2 = a 2 x 1, β 1 = b 1 b 2 x 2, β 2 =. In the case σ 2 1 = σ 2 2 = h 1 = h 2 = inequality 5.17 holds and condition 5.16 ensures asymptotic stability of the positive point of equilibrium of the classical Lotka-Volterra model 2.9. Stability regions for the positive point of equilibrium of system 2.8, obtained by conditions 5.16, 5.17, are shown in the space a, b for a 1 =.6, a 2 = 1, b 1 = 1, b 2 = 1 and different values of the other parameters on Fig. 5.1 σ 2 1 =, σ2 2 =, h 1 =, h 2 =, Fig. 5.2 σ 2 1 =.4, σ2 2 =.6, h 1 =, h 2 =, Fig. 5.3 σ 2 1 =, σ 2 2 =, h 1 =.1, h 2 =.15, Fig. 5.4 σ 2 1 =.1, σ 2 2 =.3, h 1 =.1, h 2 =.1.

14 248 L. SHAIKHET Fig.5.1 Fig.5.2 Fig.5.3 Fig.5.4

15 STABILITY OF A POSITIVE POINT OF EQUILIBRIUM SHORT SKETCH OF RESEARCH OF SYSTEM 2.1 WITH STOCHASTIC PERTURBATIONS Consider now system 2.1 that was obtained from 2.1 by conditions Via 3.1, 3.2 the positive point of equilibrium x 1, x 2 of system 2.1 is defined by the conditions 6.1 x 1 a K x 1 = K 1fx 1, x 2 x 2, b = R 1 gx 1, x 2, a > K x 1. Suppose that the functions fx 1, x 2 and gx 1, x 2 in 2.11 are differentiable and can be represented in the form fy 1 + x 1, y 2 + x 2 = f + f 1 y 1 f 2 y 2 + oy 1, y 2, gy 1 + x 1, y 2 + x 2 = g + g 1 y 1 g 2 y 2 + oy 1, y 2, where lim y oy 1,y 2 y = for y = y y2 2 and f = fx 1, x 2, f 1 = x 2 ˆf, f 2 = x 1 ˆf, ˆf = ka 2 x 1 x 2 k 1 x 1 k + a 2 x 2 k 2, g = gx 1, x 2, g 1 = x 2ĝ, g 2 = x 1ĝ, ĝ = mb 2x 1x 2 m 1 x 1 m + b 2 x 2 m 2. So, the functionals F x 1t, x 2t, F 1 x 1t, x 2t, G 1 x 1t, x 2t in 2.11 have representations 6.2 F y 1t + x 1, y 2t + x 2 = K x 1 + y 1 t sdk s, F 1 y 1t + x 1, y 2t + x 2 = K 1 f x 2 + f 1 x 2 +K 1 f y 2 t f 2 x 2 G 1 y 1t + x 1, y 2t + x 2 = R 1 g x 2 + g 1 x 2 +R 1 g y 2 t g 2 x 2 y 1 t sdk 1 s + y 2 t sdk 1 s + oy 1, y 2, y 1 t sdr 1 s + y 2 t sdr 1 s + oy 1, y 2. Via 6.1, 6.2 the linear part of system 2.1 with stochastic perturbations takes the form 6.3 ż 1 t = a K x 1z 1 t K 1 f z 2 t ż 2 t = g 1 x 2 z 1 t sdks + f 2 x 2 z 1 t sdr 1 s g 2 x 2 z 2 t sdk 1 s + σ 1 z 1 tẇ 1 t, z 2 t sdr 1 s + σ 2 z 2 tẇ 2 t,

16 25 L. SHAIKHET where dks = x 1 dk s + f 1 x 2 dk 1s. Following GMLFC one can represent system 6.3 in form 5.1 with Z 1 t = z 1 t Z 2 t = z 2 t + g 1 x 2 α 1 = K 1 x 2 f x 1 z 1 θdθdks + f 2 x 2 z 1 θdθdr 1 s g 2 x 2 z 2 θdθdk 1 s, f 1 K x 1, α 2 = K 1 f 2 x 2 f, z 2 θdθdr 1 s, β 1 = R 1 g 1 x 2, β 2 = R 1 g 2 x 2. Further investigations are similar to Section 5. For short consider system 2.13 that is a particular case of system 2.1. The point of equilibrium of system 2.13 is defined by 3.5. From 2.12, 3.5, 6.3 it follows that the linear part of system 2.13 with stochastic perturbations has the form 6.4 ż 1 t = α 1 z 1 t + α 2 z 2 t + σ 1 z 1 tẇ 1 t, ż 2 t = β 1 z 1 t h + β 2 z 2 t h + σ 2 z 2 tẇ 2 t, where α 1 = a 1 α2b + a 2 a, α 2 = B 2 a 1 α, α = 1 B + a 2 2, β 1 = b 1 b 2 β, β 2 = Bβ 1, B = bb 2 b 1 b, β = 1 B + b 2 2. System 6.4 can be represented in form 5.1 with Z 1 t = z 1 t, Z 2 t = z 2 t + β 1 z 1 θ + β 2 z 2 θdθ. Suppose that all parameters of system 2.13 are positive and besides 6.5 b 1 > b, α 1 + ɛ 1 <, β 2 + ɛ 2 <. Then α 1 + β 2 <. Since here β 1 >, α 2 < then conditions 4.7 hold. Let us show that intervals 4.8, 4.9 have common points. Really, it is easy to see that the left bounds of both intervals are nonpositive ones. Besides, α1 + β 2 2 4β 1 α 2 α 1 + β 2 2β 1 α 1 + β 2 β 1 α 1 β 1 α 1 + ɛ 1 β 1 >, i.e., at least the positive part of interval 4.8 belongs to interval 4.9. From Corollary 4.5 it follows that the trivial solution of system 6.4 with h = is asymptotically mean square stable. It means that the trivial solution of system 6.4 can be asymptotically mean square stable and for enough small h >. A coarse

17 STABILITY OF A POSITIVE POINT OF EQUILIBRIUM 251 estimate for h one can get using the functional V 1 defined in 5.4 for fixed values of µ and γ. Putting for example µ =, γ = α 2 β 1 1 >, via 6.4, 4.2 we obtain LV 1 t = 2Z 1 tα 1 z 1 t + α 2 z 2 t +2γZ 2 tβ 1 z 1 t + β 2 z 2 t + σ 2 1z 2 1t + γσ 2 2z 2 2t = 2z 1 tα 1 z 1 t + α 2 z 2 t + σ1 2 z2 1 t + γσ2 2 z2 2 t t +2γβ 1 z 1 t + β 2 z 2 t z 2 t + β 1 z 1 θ + β 2 z 2 θdθ = 2α 1 + ɛ 1 z 2 1 t + 2γβ 2 + ɛ 2 z 2 2 t +2γβ 1 +2γβ 2 β 1 z 1 tz 1 θ + β 2 z 1 tz 2 θdθ β 1 z 2 tz 1 θ + β 2 z 2 tz 2 θdθ 2α 1 + ɛ 1 z1 2 t + 2γβ 2 + ɛ 2 z2 2 [ t t +γβ 1 β 1 hz1t 2 + z1θdθ 2 +γ β 2 [β 1 hz 22 t + + β 2 hz1t 2 + z1 2 θdθ + β 2 hz 22 t + ] z2θdθ 2 z 2 2 θdθ ] 2 = [2α 1 + ɛ 1 + Q 1 h]z1 2 t + [2γβ 2 + ɛ 2 + Q 2 h]z2 2 t + Q i i=1 z 2 i θdθ, where Q 1 = γβ 1 β 1 + β 2, Q 2 = γ β 2 β 1 + β 2. Following GMLFC, the additional functional V 2 we have to choose by standard way V 2 t = 2 Q i i=1 Then for the functional V = V 1 + V 2 we have θ t + hz 2 i θdθ. LV 1 t 2[α 1 + ɛ 1 + α 2 β 1 + β 2 h]z 2 1 t +2γ[β 2 + ɛ 2 + β 2 β 1 + β 2 h]z 2 2t. As a result we obtain the following statement: if conditions 6.5 hold and α1 + ɛ 1 h < min α 2 β 1 + β 2, β 2 + ɛ 2 β 2 β 1 + β 2 then the trivial solution of system 6.4 is asymptotically mean square stable and the positive point of equilibrium of system 2.13 with stochastic perturbations stable in probability. Note that choosing optimal values of µ and γ in the functional V 1 instead of the fixed µ = and γ = α 2 β1 1 one can loosen estimation for h.

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