zbMATH — the first resource for mathematics

Local and global stability for Lotka-Volterra systems with distributed delays and instantaneous negative feedbacks. (English) Zbl 1146.34053
The paper deals with linear functional differential equations \[ x_{i}^{\prime}(t)=-\left[ b_{i}x_{i}(t)+\sum_{j=1}^{n}l_{ij}\int_{-\tau} ^{0}x_{j}(t+\theta)\,d\eta_{ij}(\theta)\right] ,\qquad i=1,2,\ldots,n, \tag{3} \] and multiple species Lotka-Volterra type models \[ x_{i}^{\prime}(t)=r_{i}(t)x_{i}(t)\left[ 1-b_{i}x_{i}(t)-\sum_{j=1}^{n} l_{ij}\int_{-\tau}^{0}x_{j}(t+\theta)\,d\eta_{ij}(\theta)\right] ,\qquad i=1,2,\ldots,n, \tag{4} \] where \(b_{i},\) \(l_{ij}\in\mathbb{R},\) \(\tau>0,\) \(r_{i}\) are positive continuous functions and \(\eta_{ij}:[-\tau,0]\rightarrow\mathbb{R}\) are normalized functions of bounded variation. If delays are sufficiently small, they can be neglected, and solutions of functional differential equations behave mainly like those of ordinary differential equations.
In fact, for a system ({4}) without nondelayed intraspecific competition (\(b_{i}=0\)), attractivity of the positive equilibrium \(x^{\ast},\) if exists, is traditionally investigated by imposing restrictions on the size of delay in intraspecific terms. However, such assumptions may not be realistic in applied problems, and the authors assume that the intraspecific competition terms \(b_{i}x_{i}(t)\) without delay dominate, in certain sense, intraspecific competition with delay and interspecific interactions.
A key problem is to establish sufficient conditions that insure diagonal dominance of the instantaneous negative feedbacks over the total competition matrix, in which case the stability of ({4}) follows irrespective of the choice of delay functions \(\eta_{ij}.\) First a criterion for the exponential stability of the linear functional differential equations ({3}) extending known results for linear equations with discrete delays is obtained. It is shown that, for \(b_{i}>0,\) the criterion is optimal and provides necessary and sufficient conditions for the stability of ({3}) in terms of the coefficients \(b_{i}\) and \(l_{ij}\) independently of the delay functions \(\eta_{ij}.\) Then new sufficient conditions for the boundedness of solutions and for the global asymptotic stability of the positive equilibrium \(x^{\ast}\) (if exists) are obtained. These also improve results reported in the literature, and even sharper conditions are derived for the system ({4}) with nondecreasing delay functions \(\eta_{ij}.\) Remarkably, the methods used in the paper do not involve Lyapunov functionals and apply to more general systems.

34K20 Stability theory of functional-differential equations
34K60 Qualitative investigation and simulation of models involving functional-differential equations
92D25 Population dynamics (general)
PDF BibTeX Cite
Full Text: DOI
[1] Campbell, S.A., Delay independent stability for additive neural networks, Differential equations dynam. systems, 9, 115-138, (2001) · Zbl 1179.34078
[2] Faria, T., Asymptotic stability for delayed logistic type equations, Math. comput. modelling, 43, 433-445, (2006) · Zbl 1145.34043
[3] Faria, T.; Liz, E., Boundedness and asymptotic stability for delayed equations of logistic type, Proc. roy. soc. Edinburgh sect. A, 133, 1057-1073, (2003) · Zbl 1056.34071
[4] Fiedler, M., Special matrices and their applications in numerical mathematics, (1986), Martinus Nijhoff Publ. (Kluwer) Dordrecht
[5] Gopalsamy, K., Stability and oscillation in delay differential equations of population dynamics, (1992), Kluwer Academic Publ. Dordrecht · Zbl 0752.34039
[6] Gopalsamy, K.; He, X.-Z., Stability in asymmetric Hopfield nets with transmission delays, Phys. D, 76, 344-358, (1994) · Zbl 0815.92001
[7] Gopalsamy, K.; He, X.-Z., Global stability in n-species competion modelled by pure-delay type systems II. non-autonomous case, Can. appl. math. Q., 6, 17-43, (1998) · Zbl 0934.34062
[8] He, X.-Z., Global stability in nonautonomous lotka – volterra systems of “pure-delay-type”, Differential integral equations, 11, 293-310, (1998) · Zbl 1022.34068
[9] Hofbauer, J.; So, J., Diagonal dominance and harmless off-diagonal delays, Proc. amer. math. soc., 128, 2675-2682, (2000) · Zbl 0952.34058
[10] Kuang, Y., Delay differential equations with applications in population dynamics, (1993), Academic Press New York · Zbl 0777.34002
[11] Kuang, Y., Global stability in delay differential systems without dominating instantaneous negative feedbacks, J. differential equations, 119, 503-532, (1995) · Zbl 0828.34066
[12] Kuang, Y., Global stability in delayed nonautonomous lotka – volterra type systems without saturated equilibrium, Differential integral equations, 9, 557-567, (1996) · Zbl 0843.34077
[13] Kuang, Y.; Smith, H.L., Global stability for infinite delay lotka – volterra type systems, J. differential equations, 103, 221-246, (1993) · Zbl 0786.34077
[14] Lu, Z.Y.; Takeuchi, Y., Permanence and global attractivity for competion lotka – volterra systems with delay, Nonlinear anal. TMA, 22, 847-856, (1994) · Zbl 0809.92025
[15] Lu, Z.Y.; Wang, W., Global stability for two-species lotka – volterra systems with delay, J. math. anal. appl., 208, 277-280, (1997) · Zbl 0874.34060
[16] Mohamad, S.; Gopalsamy, K., Exponential stability of continuous-time and discrete-time cellular neural networks with delays, Appl. math. comput., 135, 17-38, (2003) · Zbl 1030.34072
[17] Muroya, Y., Persistence and global stability in lotka – volterra delay differential systems, Appl. math. lett., 17, 795-800, (2004) · Zbl 1061.34057
[18] Saito, Y., Permanence and global stability for general lotka – volterra predator – prey systems with distributed delays, Nonlinear anal. TMA, 47, 6157-6168, (2001) · Zbl 1042.34581
[19] Saito, Y.; Hara, T.; Ma, W., Necessary and sufficient conditions for permanence and global stability of a lotka – volterra system with two delays, J. math. anal. appl., 236, 534-556, (1999) · Zbl 0944.34059
[20] Saito, Y.; Takeuchi, Y., Sharp conditions for global stability of lotka – volterra systems with delayed intraspecific competitions, Fields inst. commun., 36, 195-211, (2003) · Zbl 1162.92333
[21] Smith, H.L., Monotone dynamical systems. an introduction to the theory of competitive and cooperative systems, Math. surveys monogr., vol. 41, (1995), Amer. Math. Soc. Providence · Zbl 0821.34003
[22] So, J.W.-H.; Tang, X.H.; Zou, X., Stability in a linear delay systems without instantaneous negative feedback, SIAM J. math. anal., 33, 1297-1304, (2002) · Zbl 1019.34074
[23] So, J.W.-H.; Tang, X.H.; Zou, X., Global attractivity for non-autonomous linear delay systems, Funkcial. ekvac., 47, 25-40, (2004) · Zbl 1122.34342
[24] Tang, X.H.; Zou, X., 3/2-type criteria for global attractivity of lotka – volterra competition system without instantaneous negative feedbacks, J. differential equations, 186, 420-439, (2002) · Zbl 1028.34070
[25] Tang, X.H.; Zou, X., Global attractivity of non-autonomous lotka – volterra competition system without instantaneous negative feedback, J. differential equations, 192, 502-535, (2003) · Zbl 1035.34085
[26] van den Driessche, P.; Wu, J.; Zou, X., Stabilization role of inhibitory self-connections in a delayed neural network, Phys. D, 150, 84-90, (2001) · Zbl 1007.34072
[27] van den Driessche, P.; Zou, X., Global stability in delayed Hopfield neural network models, SIAM J. appl. math., 58, 1878-1890, (1998) · Zbl 0917.34036
[28] Wright, E.M., A non-linear difference-differential equation, J. reine angew. math., 194, 66-87, (1955) · Zbl 0064.34203
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.