STABILITY OF A POSITIVE POINT OF EQUILIBRIUM OF ONE NONLINEAR SYSTEM WITH AFTEREFFECT AND STOCHASTIC PERTURBATIONS
|
|
- Edgar Summers
- 5 years ago
- Views:
Transcription
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.
18 252 L. SHAIKHET REFERENCES [1] M. Bandyopadhyay and J. Chattopadhyay, Ratio dependent predator-prey model: effect of environmental fluctuation and stability, Nonlinearity, 18: , 25. [2] E. Beretta, V. Kolmanovskii and L. Shaikhet, Stability of epidemic model with time delays influenced by stochastic perturbations, Mathematics and Computers in Simulation Special Issue Delay Systems, 45N.3-4: , [3] E. Beretta and Y. Kuang, Global analysis in some delayed ratio-dependent predator-prey systems, Nonlinear Anal., 32N.4:381 48, [4] E. Beretta and Y. Takeuchi, Qualitative properties of chemostat equations with time delays: Boundedness, local and global asymptotic stability, Diff. Eqns and Dynamical Systems, 2N.1:19 4, [5] E. Beretta and Y. Takeuchi, Qualitative properties of chemostat equations with time delays II, Diff. Eqns and Dynamical Systems, 2N.4: , [6] E. Beretta and Y. Takeuchi, Global stability of an SIR epidemic model with time delays, J. Math. Biol, 33:25 26, [7] A.W. Bush and A.E. Cook, The effect of time delay and growth rate inhibition in the bacterial treatment of wastewater, Journal of Theoretical Biology, 63: , [8] M. Carletti, On the stability properties of a stochastic model for phage-bacteria interaction in open marine environment, Mathematical Biosciences, 175: , 22. [9] F. Chen, Periodicity in a ratio-dependent predator-prey system with stage structure for predator, Journal of Applied Mathematics, 25N.2: , 25. [1] J.B. Colliugs, The effects of the functional response on the bifurcation behavior of a mite predator-prey interaction model, J. Math. Biol, 36: , [11] M. Fan, Q. Wang and X. Zhou, Dynamics of a nonautonomous ratio-dependent predator-prey system, Proc. Royal Soc. Edin. A., 133:97 118, 23. [12] Y.H. Fan, W.T. Li and L.L. Wang, Periodic solutions of delayed ratio-dependent predatorprey model with monotonic and nomonotonic functional response, Nonlinear Analysis RWA, 5:N.2: , 24. [13] I.I. Gikhman and A.V. Skorokhod, The theory of stochastic processes, I, II, III, Springer Verlag, Berlin, 1974, 1975, [14] K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic Publishers, Dordrecht, [15] S.A. Gourley and Y. Kuang, A stage structured predator-prey model and its dependence on maturation delay and death rate, Journal of Mathematical Biology, 49:188 2, 24. [16] A. Hastings, Age dependent predation is not a simple process. I. Continuous time models, Theoretical Population Biology, 23: , [17] A. Hastings, Delays in recruitment at different trophic levels effects on stability, Journal of Mathematical Biology, 21:35 44, [18] H.F. Huo and W.T. Li, Periodic solution of a delayed predator-prey system with Michaelis- Menten type functional response, J. Comput. Appl. Math., 166: , 24. [19] V.B. Kolmanovskii and A.D. Myshkis, Applied Theory of Functional Differential Equations, Kluwer Academic Publishers, Boston, USA, [2] V.B. Kolmanovskii and L.E. Shaikhet, New results in stability theory for stochastic functionaldifferential equations SFDEs and their applications, In Proceedings of Dynamic Systems and Applications, Dynamic Publishers Inc, I: , 1994.
19 STABILITY OF A POSITIVE POINT OF EQUILIBRIUM 253 [21] V.B. Kolmanovskii and L.E. Shaikhet, A method for constructing Lyapunov functionals for stochastic differential equations of neutral type, Differentialniye uravneniya, 31N.ll: , 1995 in Russian. Translation in Differential Equations, 31N.ll: , [22] V.B. Kolmanovskii and L.E. Shaikhet, Some peculiarities of the general method of Lyapunov functionals construction, Applied Math. Lett., 15N.3:355 36, 22. [23] V.B. Kolmanovskii and L.E. Shaikhet, Construction of Lyapunov functionals for stochastic hereditary systems: a survey of some recent results, Mathematical and Computer Modelling, 36N.6: , 22. [24] V.B. Kolmanovskii and L.E. Shaikhet, About one application of the general method of Lyapunov functionals construction, International Journal of Robust and Nonlinear Control, Special Issue on Time Delay Systems, RNC., 13N.9:85 818, 23. [25] Y. Kuang, Delay differential equations with application in population dynamics, Academic Press, New York, USA, [26] P.H. Leslie, A stochastic model for studying the properties of certain biological systems by numerical methods, Biometrika, 45:16 31, [27] Y.K. Li and Y. Kuang, Periodic solutions of periodic delay Lotka-Volterra equations and systems, J. Math. Anal. Appl., 255:26 28, 21. [28] S. Ruan and D. Xiao, Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 61: , 21. [29] L.E. Shaikhet, Modem state and development perspectives of Lyapunov functionals method in the stability theory of stochastic hereditary systems, Theory of Stochastic Processes, 218N.1-2: , [3] L.E. Shaikhet, Stability in probability of nonlinear stochastic systems with delay, Matematicheskiye zametki, 57N.1: , 1995 in Russian. Translation in Math. Notes, 57N.1-2:13 16, [31] L.E. Shaikhet, Stability in probability of nonlinear stochastic hereditary systems, Dynamic Systems and Applications, 4N.2:199 24, [32] L.E. Shaikhet, Stability of predator-prey model with aftereffect by stochastic perturbations, Stability and Control: Theory and Applications, 1N.1:3 13, [33] L.L. Wang and W.T. Li, Existence and global stability of positive periodic solutions of a predator-prey system with delays, Appl. Math. Comput., 146N.1: , 23. [34] L.L. Wang and W.T. Li, Periodic solutions and stability for a delayed discrete ratio-dependent predator-prey system with Holling-type functional response, Discrete Dynamics in Nature and Society, 24N.2: , 24. [35] Q. Wang, M. Fan and K. Wang, Dynamics of a class of nonautonomous semi-ratio-dependent predator-prey system with functional responses, J. Math. Anal. Appl., 278: , 23. [36] D. Xiao and S. Ruan, Multiple bifurcations in a delayed predator-prey system with nonmonotonic functional response, J. Differential Equations, 176:494 51, 21. [37] X. Zhang, L. Chen and U.A. Neumann, The stage-structured predator-prey model and optimal harvesting policy, Mathematical Biosciences, 168:21 21, 2.
Stability of epidemic model with time delays influenced by stochastic perturbations 1
Mathematics and Computers in Simulation 45 (1998) 269±277 Stability of epidemic model with time delays influenced by stochastic perturbations 1 Edoardo Beretta a,*, Vladimir Kolmanovskii b, Leonid Shaikhet
More informationPreprint núm May Stability of a stochastic model for HIV-1 dynamics within a host. L. Shaikhet, A. Korobeinikov
Preprint núm. 91 May 2014 Stability of a stochastic model for HIV-1 dynamics within a host L. Shaikhet, A. Korobeinikov STABILITY OF A STOCHASTIC MODEL FOR HIV-1 DYNAMICS WITHIN A HOST LEONID SHAIKHET
More informationExistence of Positive Periodic Solutions of Mutualism Systems with Several Delays 1
Advances in Dynamical Systems and Applications. ISSN 973-5321 Volume 1 Number 2 (26), pp. 29 217 c Research India Publications http://www.ripublication.com/adsa.htm Existence of Positive Periodic Solutions
More informationGuangzhou, P.R. China
This article was downloaded by:[luo, Jiaowan] On: 2 November 2007 Access Details: [subscription number 783643717] Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number:
More informationOn a non-autonomous stochastic Lotka-Volterra competitive system
Available online at www.isr-publications.com/jnsa J. Nonlinear Sci. Appl., 7), 399 38 Research Article Journal Homepage: www.tjnsa.com - www.isr-publications.com/jnsa On a non-autonomous stochastic Lotka-Volterra
More informationAbout Lyapunov Functionals Construction for Difference Equations with Continuous Time
Available online at www.sciencedimct.com Applied (~ Mathematics.o,=.~. o,..o.. Letters ELSEVIER Applied Mathematics Letters 17 (2004) 985-991 www.elsevier.com/locate/aml About Lyapunov Functionals Construction
More informationDYNAMICS IN 3-SPECIES PREDATOR-PREY MODELS WITH TIME DELAYS. Wei Feng
DISCRETE AND CONTINUOUS Website: www.aimsciences.org DYNAMICAL SYSTEMS SUPPLEMENT 7 pp. 36 37 DYNAMICS IN 3-SPECIES PREDATOR-PREY MODELS WITH TIME DELAYS Wei Feng Mathematics and Statistics Department
More informationPERMANENCE IN LOGISTIC AND LOTKA-VOLTERRA SYSTEMS WITH DISPERSAL AND TIME DELAYS
Electronic Journal of Differential Equations, Vol. 2005(2005), No. 60, pp. 1 11. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) PERMANENCE
More informationOSCILLATION AND ASYMPTOTIC STABILITY OF A DELAY DIFFERENTIAL EQUATION WITH RICHARD S NONLINEARITY
2004 Conference on Diff. Eqns. and Appl. in Math. Biology, Nanaimo, BC, Canada. Electronic Journal of Differential Equations, Conference 12, 2005, pp. 21 27. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu
More informationAnalysis of a predator prey model with modified Leslie Gower and Holling-type II schemes with time delay
Nonlinear Analysis: Real World Applications 7 6 4 8 www.elsevier.com/locate/na Analysis of a predator prey model with modified Leslie Gower Holling-type II schemes with time delay A.F. Nindjin a, M.A.
More informationGlobal Stability Analysis on a Predator-Prey Model with Omnivores
Applied Mathematical Sciences, Vol. 9, 215, no. 36, 1771-1782 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.12988/ams.215.512 Global Stability Analysis on a Predator-Prey Model with Omnivores Puji Andayani
More informationAbout Stability of Nonlinear Stochastic Difference Equations
PERGAMON Applied Mathematics Letters 13 (2000) 27-32 Applied Mathematics Letters www.elsevier.nl/locate/aml About Stability of Nonlinear Stochastic Difference Equations B. PATERNOSTER Dipart. Informatica
More informationGlobal Qualitative Analysis for a Ratio-Dependent Predator Prey Model with Delay 1
Journal of Mathematical Analysis and Applications 266, 401 419 (2002 doi:10.1006/jmaa.2001.7751, available online at http://www.idealibrary.com on Global Qualitative Analysis for a Ratio-Dependent Predator
More informationNonstandard Finite Difference Methods For Predator-Prey Models With General Functional Response
Nonstandard Finite Difference Methods For Predator-Prey Models With General Functional Response Dobromir T. Dimitrov Hristo V. Kojouharov Technical Report 2007-0 http://www.uta.edu/math/preprint/ NONSTANDARD
More informationBifurcation and Stability Analysis of a Prey-predator System with a Reserved Area
ISSN 746-733, England, UK World Journal of Modelling and Simulation Vol. 8 ( No. 4, pp. 85-9 Bifurcation and Stability Analysis of a Prey-predator System with a Reserved Area Debasis Mukherjee Department
More informationPersistence and global stability in discrete models of Lotka Volterra type
J. Math. Anal. Appl. 330 2007 24 33 www.elsevier.com/locate/jmaa Persistence global stability in discrete models of Lotka Volterra type Yoshiaki Muroya 1 Department of Mathematical Sciences, Waseda University,
More informationDynamics of Modified Leslie-Gower Predator-Prey Model with Predator Harvesting
International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:13 No:05 55 Dynamics of Modified Leslie-Gower Predator-Prey Model with Predator Harvesting K. Saleh Department of Mathematics, King Fahd
More informationGlobal dynamics of a predator-prey system with Holling type II functional response
4 Nonlinear Analysis: Modelling and Control, 011, Vol. 16, No., 4 53 Global dynamics of a predator-prey system with Holling type II functional response Xiaohong Tian, Rui Xu Institute of Applied Mathematics
More informationDYNAMICS IN A COMPETITIVE LOTKA-VOLTERRA PREDATOR-PREY MODEL WITH MULTIPLE DELAYS. Changjin Xu 1. Yusen Wu. 1. Introduction
italian journal of pure and applied mathematics n. 36 2016 (231 244) 231 DYNAMICS IN A COMPETITIVE LOTKA-VOLTERRA PREDATOR-PREY MODEL WITH MULTIPLE DELAYS Changjin Xu 1 Guizhou Key Laboratory of Economics
More informationStability and bifurcation in a two species predator-prey model with quintic interactions
Chaotic Modeling and Simulation (CMSIM) 4: 631 635, 2013 Stability and bifurcation in a two species predator-prey model with quintic interactions I. Kusbeyzi Aybar 1 and I. acinliyan 2 1 Department of
More informationGLOBAL ATTRACTIVITY IN A NONLINEAR DIFFERENCE EQUATION
Sixth Mississippi State Conference on ifferential Equations and Computational Simulations, Electronic Journal of ifferential Equations, Conference 15 (2007), pp. 229 238. ISSN: 1072-6691. URL: http://ejde.mathmississippi
More informationStochastic Viral Dynamics with Beddington-DeAngelis Functional Response
Stochastic Viral Dynamics with Beddington-DeAngelis Functional Response Junyi Tu, Yuncheng You University of South Florida, USA you@mail.usf.edu IMA Workshop in Memory of George R. Sell June 016 Outline
More informationON THE BEHAVIOR OF SOLUTIONS OF LINEAR NEUTRAL INTEGRODIFFERENTIAL EQUATIONS WITH UNBOUNDED DELAY
Georgian Mathematical Journal Volume 11 (24), Number 2, 337 348 ON THE BEHAVIOR OF SOLUTIONS OF LINEAR NEUTRAL INTEGRODIFFERENTIAL EQUATIONS WITH UNBOUNDED DELAY I.-G. E. KORDONIS, CH. G. PHILOS, I. K.
More informationDYNAMICS OF A PREDATOR-PREY INTERACTION IN CHEMOSTAT WITH VARIABLE YIELD
Journal of Sustainability Science Management Volume 10 Number 2, December 2015: 16-23 ISSN: 1823-8556 Penerbit UMT DYNAMICS OF A PREDATOR-PREY INTERACTION IN CHEMOSTAT WITH VARIABLE YIELD SARKER MD SOHEL
More informationStability of equilibrium states of a nonlinear delay differential equation with stochastic perturbations
INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL Int. J. Robust Nonlinear Control (2016) Published online in Wiley Online Library (wileyonlinelibrary.com)..3605 Stability of equilibrium states of
More informationSTUDY OF THE DYNAMICAL MODEL OF HIV
STUDY OF THE DYNAMICAL MODEL OF HIV M.A. Lapshova, E.A. Shchepakina Samara National Research University, Samara, Russia Abstract. The paper is devoted to the study of the dynamical model of HIV. An application
More informationPermanency and Asymptotic Behavior of The Generalized Lotka-Volterra Food Chain System
CJMS. 5(1)(2016), 1-5 Caspian Journal of Mathematical Sciences (CJMS) University of Mazandaran, Iran http://cjms.journals.umz.ac.ir ISSN: 1735-0611 Permanency and Asymptotic Behavior of The Generalized
More informationNONSTANDARD NUMERICAL METHODS FOR A CLASS OF PREDATOR-PREY MODELS WITH PREDATOR INTERFERENCE
Sixth Mississippi State Conference on Differential Equations and Computational Simulations, Electronic Journal of Differential Equations, Conference 15 (2007), pp. 67 75. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu
More informationSTABILITY ANALYSIS OF DELAY DIFFERENTIAL EQUATIONS WITH TWO DISCRETE DELAYS
CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 20, Number 4, Winter 2012 STABILITY ANALYSIS OF DELAY DIFFERENTIAL EQUATIONS WITH TWO DISCRETE DELAYS XIHUI LIN AND HAO WANG ABSTRACT. We use an algebraic
More informationPERSISTENCE AND PERMANENCE OF DELAY DIFFERENTIAL EQUATIONS IN BIOMATHEMATICS
PERSISTENCE AND PERMANENCE OF DELAY DIFFERENTIAL EQUATIONS IN BIOMATHEMATICS PhD Thesis by Nahed Abdelfattah Mohamady Abdelfattah Supervisors: Prof. István Győri Prof. Ferenc Hartung UNIVERSITY OF PANNONIA
More informationGlobal Attractivity in a Nonlinear Difference Equation and Applications to a Biological Model
International Journal of Difference Equations ISSN 0973-6069, Volume 9, Number 2, pp. 233 242 (204) http://campus.mst.edu/ijde Global Attractivity in a Nonlinear Difference Equation and Applications to
More informationAnalysis of a delayed predator-prey model with ratio-dependent functional response and quadratic harvesting. Peng Feng
Analysis of a delayed predator-prey model with ratio-dependent functional response and quadratic harvesting Peng Feng Journal of Applied Mathematics and Computing ISSN 1598-5865 J. Appl. Math. Comput.
More informationHOPF BIFURCATION ANALYSIS OF A PREDATOR-PREY SYSTEM WITH NON-SELECTIVE HARVESTING AND TIME DELAY
Vol. 37 17 No. J. of Math. PRC HOPF BIFURCATION ANALYSIS OF A PREDATOR-PREY SYSTEM WITH NON-SELECTIVE HARVESTING AND TIME DELAY LI Zhen-wei, LI Bi-wen, LIU Wei, WANG Gan School of Mathematics and Statistics,
More informationHIGHER ORDER CRITERION FOR THE NONEXISTENCE OF FORMAL FIRST INTEGRAL FOR NONLINEAR SYSTEMS. 1. Introduction
Electronic Journal of Differential Equations, Vol. 217 (217), No. 274, pp. 1 11. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu HIGHER ORDER CRITERION FOR THE NONEXISTENCE
More informationA Comparison of Two Predator-Prey Models with Holling s Type I Functional Response
A Comparison of Two Predator-Prey Models with Holling s Type I Functional Response ** Joint work with Mark Kot at the University of Washington ** Mathematical Biosciences 1 (8) 161-179 Presented by Gunog
More informationLOCAL AND GLOBAL STABILITY OF IMPULSIVE PEST MANAGEMENT MODEL WITH BIOLOGICAL HYBRID CONTROL
International J. of Math. Sci. & Engg. Appls. (IJMSEA) ISSN 0973-9424, Vol. 11 No. II (August, 2017), pp. 129-141 LOCAL AND GLOBAL STABILITY OF IMPULSIVE PEST MANAGEMENT MODEL WITH BIOLOGICAL HYBRID CONTROL
More informationStability Analysis And Maximum Profit Of Logistic Population Model With Time Delay And Constant Effort Of Harvesting
Jurnal Vol 3, Matematika, No, 9-8, Juli Statistika 006 & Komputasi Vol No Juli 006 9-8, Juli 006 9 Stability Analysis And Maximum Profit Of Logistic Population Model With Time Delay And Constant Effort
More informationVITA EDUCATION. Arizona State University, , Ph.D. Huazhong Univ. of Sci. and Tech., , M.S. Lanzhou University, , B.S.
VITA Bingtuan Li Department of Mathematics University of Louisville Louisville, KY 40292 Office phone: (502) 852-6149 Fax: (502) 852-7132 E-mail: bing.li@louisville.edu EDUCATION Arizona State University,
More informationGlobal attractivity and positive almost periodic solution of a multispecies discrete mutualism system with time delays
Global attractivity and positive almost periodic solution of a multispecies discrete mutualism system with time delays Hui Zhang Abstract In this paper, we consider an almost periodic multispecies discrete
More informationResearch Article On the Stability Property of the Infection-Free Equilibrium of a Viral Infection Model
Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume, Article ID 644, 9 pages doi:.55//644 Research Article On the Stability Property of the Infection-Free Equilibrium of a Viral
More informationPermanence Implies the Existence of Interior Periodic Solutions for FDEs
International Journal of Qualitative Theory of Differential Equations and Applications Vol. 2, No. 1 (2008), pp. 125 137 Permanence Implies the Existence of Interior Periodic Solutions for FDEs Xiao-Qiang
More informationGlobal Attractivity in a Higher Order Difference Equation with Applications
International Jr, of Qualitative Theory of Differential Equations and Applications International Vol. 3 No. 1 Jr, (January-June, of Qualitative 2017) Theory of Differential Equations and Applications Vol.
More informationThe exponential asymptotic stability of singularly perturbed delay differential equations with a bounded lag
J. Math. Anal. Appl. 270 (2002) 143 149 www.academicpress.com The exponential asymptotic stability of singularly perturbed delay differential equations with a bounded lag Hongjiong Tian Department of Mathematics,
More informationSome Integral Inequalities with Maximum of the Unknown Functions
Advances in Dynamical Systems and Applications ISSN 0973-5321, Volume 6, Number 1, pp. 57 69 2011 http://campus.mst.edu/adsa Some Integral Inequalities with Maximum of the Unknown Functions Snezhana G.
More informationStability and Hopf bifurcation of a ratio-dependent predator prey model with time delay and stage structure
Electronic Journal of Qualitative Theory of Differential Equations 6, No 99, 3; doi: 43/ejqtde699 http://wwwmathu-szegedhu/ejqtde/ Stability and Hopf bifurcation of a ratio-dependent predator prey model
More informationESAIM: M2AN Modélisation Mathématique et Analyse Numérique M2AN, Vol. 37, N o 2, 2003, pp DOI: /m2an:
Mathematical Modelling and Numerical Analysis ESAIM: M2AN Modélisation Mathématique et Analyse Numérique M2AN, Vol. 37, N o 2, 2003, pp. 339 344 DOI: 10.1051/m2an:2003029 PERSISTENCE AND BIFURCATION ANALYSIS
More informationStability Theory for Nonnegative and Compartmental Dynamical Systems with Time Delay
1 Stability Theory for Nonnegative and Compartmental Dynamical Systems with Time Delay Wassim M. Haddad and VijaySekhar Chellaboina School of Aerospace Engineering, Georgia Institute of Technology, Atlanta,
More informationASTATISM IN NONLINEAR CONTROL SYSTEMS WITH APPLICATION TO ROBOTICS
dx dt DIFFERENTIAL EQUATIONS AND CONTROL PROCESSES N 1, 1997 Electronic Journal, reg. N P23275 at 07.03.97 http://www.neva.ru/journal e-mail: diff@osipenko.stu.neva.ru Control problems in nonlinear systems
More informationHarvesting Model for Fishery Resource with Reserve Area and Modified Effort Function
Malaya J. Mat. 4(2)(2016) 255 262 Harvesting Model for Fishery Resource with Reserve Area and Modified Effort Function Bhanu Gupta and Amit Sharma P.G. Department of Mathematics, JC DAV College, Dasuya
More information2D-Volterra-Lotka Modeling For 2 Species
Majalat Al-Ulum Al-Insaniya wat - Tatbiqiya 2D-Volterra-Lotka Modeling For 2 Species Alhashmi Darah 1 University of Almergeb Department of Mathematics Faculty of Science Zliten Libya. Abstract The purpose
More informationA NONLINEAR NEUTRAL PERIODIC DIFFERENTIAL EQUATION
Electronic Journal of Differential Equations, Vol. 2010(2010), No. 88, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu A NONLINEAR NEUTRAL
More informationStationary distribution and pathwise estimation of n-species mutualism system with stochastic perturbation
Available online at www.tjnsa.com J. Nonlinear Sci. Appl. 9 6), 936 93 Research Article Stationary distribution and pathwise estimation of n-species mutualism system with stochastic perturbation Weiwei
More informationTwo Modifications of the Beverton Holt Equation
International Journal of Difference Equations ISSN 0973-5321 Volume 4 Number 1 pp. 115 136 2009 http://campus.mst.edu/ijde Two Modifications of the Beverton Holt Equation G. Papaschinopoulos. J. Schinas
More informationDYNAMICS OF A RATIO-DEPENDENT PREY-PREDATOR SYSTEM WITH SELECTIVE HARVESTING OF PREDATOR SPECIES. 1. Introduction
J. Appl. Math. & Computing Vol. 232007), No. 1-2, pp. 385-395 Website: http://jamc.net DYNAMICS OF A RATIO-DEPENDENT PREY-PREDATOR SYSTEM WITH SELECTIVE HARVESTING OF PREDATOR SPECIES TAPAN KUMAR KAR Abstract.
More informationA Stochastic Viral Infection Model with General Functional Response
Nonlinear Analysis and Differential Equations, Vol. 4, 16, no. 9, 435-445 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.1988/nade.16.664 A Stochastic Viral Infection Model with General Functional Response
More informationOn predator-prey systems and small-gain theorems
On predator-prey systems and small-gain theorems Patrick De Leenheer, David Angeli and Eduardo D. Sontag December 4, Abstract This paper deals with an almost global attractivity result for Lotka-Volterra
More informationON THE ASYMPTOTIC STABILITY IN TERMS OF TWO MEASURES FOR FUNCTIONAL DIFFERENTIAL EQUATIONS. G. Makay
ON THE ASYMPTOTIC STABILITY IN TERMS OF TWO MEASURES FOR FUNCTIONAL DIFFERENTIAL EQUATIONS G. Makay Student in Mathematics, University of Szeged, Szeged, H-6726, Hungary Key words and phrases: Lyapunov
More informationDiscrete Population Models with Asymptotically Constant or Periodic Solutions
International Journal of Difference Equations ISSN 0973-6069, Volume 6, Number 2, pp. 143 152 (2011) http://campus.mst.edu/ijde Discrete Population Models with Asymptotically Constant or Periodic Solutions
More informationChaos and adaptive control in two prey, one predator system with nonlinear feedback
Chaos and adaptive control in two prey, one predator system with nonlinear feedback Awad El-Gohary, a, and A.S. Al-Ruzaiza a a Department of Statistics and O.R., College of Science, King Saud University,
More informationx 2 F 1 = 0 K 2 v 2 E 1 E 2 F 2 = 0 v 1 K 1 x 1
ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 20, Number 4, Fall 1990 ON THE STABILITY OF ONE-PREDATOR TWO-PREY SYSTEMS M. FARKAS 1. Introduction. The MacArthur-Rosenzweig graphical criterion" of stability
More informationDISSIPATION CONTROL OF AN N-SPECIES FOOD CHAIN SYSTEM
INTERNATIONAL JOURNAL OF INFORMATION AND SYSTEMS SCIENCES Volume 1 Number 3-4 Pages 48 440 c 005 Institute for Scientific Computing and Information DISSIPATION CONTROL OF AN N-SPECIES FOOD CHAIN SYSTEM
More informationFeedback-mediated oscillatory coexistence in the chemostat
Feedback-mediated oscillatory coexistence in the chemostat Patrick De Leenheer and Sergei S. Pilyugin Department of Mathematics, University of Florida deleenhe,pilyugin@math.ufl.edu 1 Introduction We study
More informationThe Application of the Poincart-Transform to the Lotka-Volterra Model
J. Math. Biology 6, 67-73 (1978) Journal of by Springer-Verlag 1978 The Application of the Poincart-Transform to the Lotka-Volterra Model S. B. Hsu Department of Mathematics, University of Utah, Salt Lake
More informationGLOBAL ATTRACTIVITY IN A CLASS OF NONMONOTONE REACTION-DIFFUSION EQUATIONS WITH TIME DELAY
CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 17, Number 1, Spring 2009 GLOBAL ATTRACTIVITY IN A CLASS OF NONMONOTONE REACTION-DIFFUSION EQUATIONS WITH TIME DELAY XIAO-QIANG ZHAO ABSTRACT. The global attractivity
More informationStability of solutions to abstract evolution equations with delay
Stability of solutions to abstract evolution equations with delay A.G. Ramm Department of Mathematics Kansas State University, Manhattan, KS 66506-2602, USA ramm@math.ksu.edu Abstract An equation u = A(t)u+B(t)F
More informationAsymptotic stability of solutions of a class of neutral differential equations with multiple deviating arguments
Bull. Math. Soc. Sci. Math. Roumanie Tome 57(15) No. 1, 14, 11 13 Asymptotic stability of solutions of a class of neutral differential equations with multiple deviating arguments by Cemil Tunç Abstract
More informationEXISTENCE OF POSITIVE PERIODIC SOLUTIONS OF DISCRETE MODEL FOR THE INTERACTION OF DEMAND AND SUPPLY. S. H. Saker
Nonlinear Funct. Anal. & Appl. Vol. 10 No. 005 pp. 311 34 EXISTENCE OF POSITIVE PERIODIC SOLUTIONS OF DISCRETE MODEL FOR THE INTERACTION OF DEMAND AND SUPPLY S. H. Saker Abstract. In this paper we derive
More informationOBSERVABILITY AND OBSERVERS IN A FOOD WEB
OBSERVABILITY AND OBSERVERS IN A FOOD WEB I. LÓPEZ, M. GÁMEZ, AND S. MOLNÁR Abstract. The problem of the possibility to recover the time-dependent state of a whole population system out of the observation
More informationEXISTENCE OF NEUTRAL STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS WITH ABSTRACT VOLTERRA OPERATORS
International Journal of Differential Equations and Applications Volume 7 No. 1 23, 11-17 EXISTENCE OF NEUTRAL STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS WITH ABSTRACT VOLTERRA OPERATORS Zephyrinus C.
More informationDynamics on a General Stage Structured n Parallel Food Chains
Memorial University of Newfoundland Dynamics on a General Stage Structured n Parallel Food Chains Isam Al-Darabsah and Yuan Yuan November 4, 2013 Outline: Propose a general model with n parallel food chains
More informationApplied Mathematics Letters
PERGAMON Applied Matheatics Letters 15 (2002) 355-360 Applied Matheatics Letters www.elsevier.co/locate/al Soe Peculiarities of the General Method of Lyapunov Functionals Construction V. KOLMANOVSKII Moscow
More informationASYMPTOTIC BEHAVIOR OF SOLUTIONS OF DISCRETE VOLTERRA EQUATIONS. Janusz Migda and Małgorzata Migda
Opuscula Math. 36, no. 2 (2016), 265 278 http://dx.doi.org/10.7494/opmath.2016.36.2.265 Opuscula Mathematica ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF DISCRETE VOLTERRA EQUATIONS Janusz Migda and Małgorzata
More informationBifurcation Analysis of Prey-Predator Model with Harvested Predator
International Journal of Engineering Research and Development e-issn: 78-67X, p-issn: 78-8X, www.ijerd.com Volume, Issue 6 (June 4), PP.4-5 Bifurcation Analysis of Prey-Predator Model with Harvested Predator
More informationEVANESCENT SOLUTIONS FOR LINEAR ORDINARY DIFFERENTIAL EQUATIONS
EVANESCENT SOLUTIONS FOR LINEAR ORDINARY DIFFERENTIAL EQUATIONS Cezar Avramescu Abstract The problem of existence of the solutions for ordinary differential equations vanishing at ± is considered. AMS
More informationBIBO stabilization of feedback control systems with time dependent delays
to appear in Applied Mathematics and Computation BIBO stabilization of feedback control systems with time dependent delays Essam Awwad a,b, István Győri a and Ferenc Hartung a a Department of Mathematics,
More informationResearch Article The Mathematical Study of Pest Management Strategy
Discrete Dynamics in Nature and Society Volume 22, Article ID 25942, 9 pages doi:.55/22/25942 Research Article The Mathematical Study of Pest Management Strategy Jinbo Fu and Yanzhen Wang Minnan Science
More informationLYAPUNOV-RAZUMIKHIN METHOD FOR DIFFERENTIAL EQUATIONS WITH PIECEWISE CONSTANT ARGUMENT. Marat Akhmet. Duygu Aruğaslan
DISCRETE AND CONTINUOUS doi:10.3934/dcds.2009.25.457 DYNAMICAL SYSTEMS Volume 25, Number 2, October 2009 pp. 457 466 LYAPUNOV-RAZUMIKHIN METHOD FOR DIFFERENTIAL EQUATIONS WITH PIECEWISE CONSTANT ARGUMENT
More informationLOTKA-VOLTERRA SYSTEMS WITH DELAY
870 1994 133-140 133 LOTKA-VOLTERRA SYSTEMS WITH DELAY Zhengyi LU and Yasuhiro TAKEUCHI Department of Applied Mathematics, Faculty of Engineering, Shizuoka University, Hamamatsu 432, JAPAN ABSTRACT Sufftcient
More informationBoundedness and asymptotic stability for delayed equations of logistic type
Proceedings of the Royal Society of Edinburgh, 133A, 1057{1073, 2003 Boundedness and asymptotic stability for delayed equations of logistic type Teresa Faria Departamento de Matem atica, Faculdade de Ci^encias/CMAF,
More informationExistence of Almost Periodic Solutions of Discrete Ricker Delay Models
International Journal of Difference Equations ISSN 0973-6069, Volume 9, Number 2, pp. 187 205 (2014) http://campus.mst.edu/ijde Existence of Almost Periodic Solutions of Discrete Ricker Delay Models Yoshihiro
More informationStability of efficient solutions for semi-infinite vector optimization problems
Stability of efficient solutions for semi-infinite vector optimization problems Z. Y. Peng, J. T. Zhou February 6, 2016 Abstract This paper is devoted to the study of the stability of efficient solutions
More informationLotka Volterra Predator-Prey Model with a Predating Scavenger
Lotka Volterra Predator-Prey Model with a Predating Scavenger Monica Pescitelli Georgia College December 13, 2013 Abstract The classic Lotka Volterra equations are used to model the population dynamics
More informationDynamics of an almost periodic facultative mutualism model with time delays
Available online at www.tjnsa.com J. Nonlinear Sci. Appl. 9 (06), 36 330 Research Article Dynamics of an almost periodic facultative mutualism model with time delays Zunguang Guo, Can Li Department of
More informationBIOCOMP 11 Stability Analysis of Hybrid Stochastic Gene Regulatory Networks
BIOCOMP 11 Stability Analysis of Hybrid Stochastic Gene Regulatory Networks Anke Meyer-Baese, Claudia Plant a, Jan Krumsiek, Fabian Theis b, Marc R. Emmett c and Charles A. Conrad d a Department of Scientific
More informationLOCAL STABILITY IMPLIES GLOBAL STABILITY IN SOME ONE-DIMENSIONAL DISCRETE SINGLE-SPECIES MODELS. Eduardo Liz. (Communicated by Linda Allen)
DISCRETE AND CONTINUOUS Website: http://aimsciences.org DYNAMICAL SYSTEMS SERIES B Volume 7, Number 1, January 2007 pp. 191 199 LOCAL STABILITY IMPLIES GLOBAL STABILITY IN SOME ONE-DIMENSIONAL DISCRETE
More informationGrowth models for cells in the chemostat
Growth models for cells in the chemostat V. Lemesle, J-L. Gouzé COMORE Project, INRIA Sophia Antipolis BP93, 06902 Sophia Antipolis, FRANCE Valerie.Lemesle, Jean-Luc.Gouze@sophia.inria.fr Abstract A chemostat
More informationPermanence and global stability of a May cooperative system with strong and weak cooperative partners
Zhao et al. Advances in Difference Equations 08 08:7 https://doi.org/0.86/s366-08-68-5 R E S E A R C H Open Access ermanence and global stability of a May cooperative system with strong and weak cooperative
More informationNONTRIVIAL SOLUTIONS TO INTEGRAL AND DIFFERENTIAL EQUATIONS
Fixed Point Theory, Volume 9, No. 1, 28, 3-16 http://www.math.ubbcluj.ro/ nodeacj/sfptcj.html NONTRIVIAL SOLUTIONS TO INTEGRAL AND DIFFERENTIAL EQUATIONS GIOVANNI ANELLO Department of Mathematics University
More informationHigher Order Averaging : periodic solutions, linear systems and an application
Higher Order Averaging : periodic solutions, linear systems and an application Hartono and A.H.P. van der Burgh Faculty of Information Technology and Systems, Department of Applied Mathematical Analysis,
More informationOscillationofNonlinearFirstOrderNeutral Di erenceequations
AppliedMathematics E-Notes, 1(2001), 5-10 c Availablefreeatmirrorsites ofhttp://math2.math.nthu.edu.tw/»amen/ OscillationofNonlinearFirstOrderNeutral Di erenceequations YingGaoandGuangZhang yz Received1June2000
More informationOptimal harvesting policy of a stochastic delay predator-prey model with Lévy jumps
Available online at www.isr-publications.com/jnsa J. Nonlinear Sci. Appl., 1 (217), 4222 423 Research Article Journal Homepage: www.tjnsa.com - www.isr-publications.com/jnsa Optimal harvesting policy of
More informationApplied Mathematics Letters. Stationary distribution, ergodicity and extinction of a stochastic generalized logistic system
Applied Mathematics Letters 5 (1) 198 1985 Contents lists available at SciVerse ScienceDirect Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml Stationary distribution, ergodicity
More informationOn Stability of Dynamic Equations on Time Scales Via Dichotomic Maps
Available at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 93-9466 Vol. 7, Issue (December ), pp. 5-57 Applications and Applied Mathematics: An International Journal (AAM) On Stability of Dynamic Equations
More informationCONVERGENCE OF SOLUTIONS OF CERTAIN NON-HOMOGENEOUS THIRD ORDER ORDINARY DIFFERENTIAL EQUATIONS
Kragujevac J. Math. 31 (2008) 5 16. CONVERGENCE OF SOLUTIONS OF CERTAIN NON-HOMOGENEOUS THIRD ORDER ORDINARY DIFFERENTIAL EQUATIONS A. U. Afuwapẹ 1 and M.O. Omeike 2 1 Departmento de Matemáticas, Universidad
More informationKey words and phrases. Bifurcation, Difference Equations, Fixed Points, Predator - Prey System, Stability.
ISO 9001:008 Certified Volume, Issue, March 013 Dynamical Behavior in a Discrete Prey- Predator Interactions M.ReniSagaya Raj 1, A.George Maria Selvam, R.Janagaraj 3.and D.Pushparajan 4 1,,3 Sacred Heart
More informationCDS Solutions to the Midterm Exam
CDS 22 - Solutions to the Midterm Exam Instructor: Danielle C. Tarraf November 6, 27 Problem (a) Recall that the H norm of a transfer function is time-delay invariant. Hence: ( ) Ĝ(s) = s + a = sup /2
More informationConvex Optimization Approach to Dynamic Output Feedback Control for Delay Differential Systems of Neutral Type 1,2
journal of optimization theory and applications: Vol. 127 No. 2 pp. 411 423 November 2005 ( 2005) DOI: 10.1007/s10957-005-6552-7 Convex Optimization Approach to Dynamic Output Feedback Control for Delay
More informationGENERALIZATION OF GRONWALL S INEQUALITY AND ITS APPLICATIONS IN FUNCTIONAL DIFFERENTIAL EQUATIONS
Communications in Applied Analysis 19 (215), 679 688 GENERALIZATION OF GRONWALL S INEQUALITY AND ITS APPLICATIONS IN FUNCTIONAL DIFFERENTIAL EQUATIONS TINGXIU WANG Department of Mathematics, Texas A&M
More informationGlobal Stability of SEIRS Models in Epidemiology
Global Stability of SRS Models in pidemiology M. Y. Li, J. S. Muldowney, and P. van den Driessche Department of Mathematics and Statistics Mississippi State University, Mississippi State, MS 39762 Department
More informationMultiple focus and Hopf bifurcations in a predator prey system with nonmonotonic functional response
Multiple focus and Hopf bifurcations in a predator prey system with nonmonotonic functional response Dongmei Xiao Department of Mathematics Shanghai Jiao Tong University, Shanghai 200030, China Email:
More information