×

zbMATH — the first resource for mathematics

Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation. (English) Zbl 1190.34064
Consider the predator-prey models
\[ dx = [a x - b x^2 - c_1 \varphi (x,y) x] dt + \alpha x dW_1, \]
\[ dy = [r y - c_2 \varphi (x,y) y] dt + \beta y dW_2, \] where \(x\) and \(y\) represent the population densities of prey and predator at time \(t \geq 0\); \(a\) and \(r\), \(b\), and \(c_i\) are positive constants of prey and predator intrinsic growth rates, death rate measuring the strength of competition among the prey individuals and conversion rates of the predator-prey model, respectively. Moreover, \(\varphi (x,y) \) models the functional response of the predator to the prey density per unit time.
In this paper, the authors are interested in the study of the case of modified Leslie-Gower and Holling-type-II functionals of the form
\[ \varphi (x,y) = y / (m+x), \] where \(m\geq 0\) measures the extent to which the environment provides protection to the prey \(x\). Now, suppose these models are perturbed by independent standard Wiener processes \(W_i\) in the parameters \(a\) and \(b\) with variance parameters \(\alpha\) and \(\beta\). Thus, one arrives at nonlinear stochastic differential equations with linear multiplicative noise and quadratic nonlinearities.
First, the authors prove the existence of unique, positive global solutions to these systems with positive initial values \(x_0, y_0 > 0\). Second, they study its long-time behavior. Third, the system is investigated with respect to extinction and persistence conditions. As their major proof technique, the authors apply well-known comparison theorems on upper and lower solution bounds and the knowledge on explicit solutions of stochastic logistic equations with linear multiplicative white noise in \(\mathbb{R}^1\). Finally, standard Milstein methods are used for some numerical simulations to illustrate their findings.

MSC:
34F05 Ordinary differential equations and systems with randomness
60H30 Applications of stochastic analysis (to PDEs, etc.)
34C60 Qualitative investigation and simulation of ordinary differential equation models
34D05 Asymptotic properties of solutions to ordinary differential equations
37H10 Generation, random and stochastic difference and differential equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
92D25 Population dynamics (general)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Arnold, L., Stochastic differential equations: theory and applications, (1972), Wiley New York
[2] Arnold, L.; Horsthemke, W.; Stucki, J.W., The influence of external real and white noise on the lotka – volterra model, Biomedical J., 21, 451-471, (1979) · Zbl 0433.92019
[3] Aziz-Alaoui, M.A.; Daher Okiye, M., Boundedness and global stability for a predator – prey model with modified leslie – gower and Holling-type II schemes, Appl. math. lett., 16, 1069-1075, (2003) · Zbl 1063.34044
[4] Bandyopadhyay, M.; Chattopadhyay, J., Ratio-dependent predator – prey model: effect of environmental fluctuation and stability, Nonlinearity, 18, 913-936, (2005) · Zbl 1078.34035
[5] Chen, L.; Chen, J., Nonlinear biological dynamical system, (1993), Science Press Beijing
[6] Chesson, P.L.; Warner, R.R., Environmental variability promotes coexistence in lottery competitive systems, Amer. natur., 117, 923-943, (1981)
[7] Du, N.H.; Sam, V.H., Dynamics of a stochastic lotka – volterra model perturbed by white noise, J. math. anal. appl., 324, 82-97, (2006) · Zbl 1107.92038
[8] Freedman, H.I.; Waltman, P., Periodic solutions of perturbed lotka – volterra systems, (), 312-316 · Zbl 0337.92015
[9] Freedman, H.I., Deterministic mathematical models in population ecology, (1980), Marcel Dekker New York · Zbl 0448.92023
[10] Friedman, A., Stochastic differential equations and their applications, (1976), Academic Press New York
[11] Klebaner, F.C., Introduction to stochastic calculus with applications, (1998), Imperial College Press · Zbl 0926.60002
[12] Guo, H.; Song, X., An impulsive predator – prey system with modified leslie – gower and Holling type II schemes, Chaos solitons fractals, 36, 1320-1331, (2008) · Zbl 1148.34034
[13] Higham, D.J., An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM rev., 43, 525-546, (2001) · Zbl 0979.65007
[14] Holling, C.S., The functional response of predator to prey density and its role in mimicry and population regulation, Men. ent. sec. can., 45, 1-60, (1965)
[15] Ikeda, N.; Wantanabe, S., Stochastic differential equations and diffusion processes, (1981), North-Holland Amsterdam
[16] Jiang, G.; Lu, Q.; Qian, L., Complex dynamics of a Holling type II prey – predator system with state feedback control, Chaos solitons fractals, 31, 448-461, (2007) · Zbl 1203.34071
[17] Leslie, P.H., Some further notes on the use of matrices in population mathematic, Biometrica, 35, 213-245, (1948) · Zbl 0034.23303
[18] Leslie, P.H.; Gower, J.C., The properties of a stochastic model for the predator – prey type of interaction between two species, Biometrica, 47, 219-234, (1960) · Zbl 0103.12502
[19] Levin, A., Dispersion and population interactions, Amer. natur., 108, 207-228, (1974)
[20] Liu, B.; Teng, Z.; Chen, L., Analysis of a predator – prey model with Holling II functional response concerning impulsive control strategy, J. comput. appl. math., 193, 347-362, (2006) · Zbl 1089.92060
[21] Liu, X.; Chen, L., Complex dynamics of Holling type II lotka – volterra predator – prey system with impulsive perturbations on the predator, Chaos solitons fractals, 16, 311-320, (2003) · Zbl 1085.34529
[22] Mao, X., Stochastic differential equations and applications, (1997), Horwood New York · Zbl 0874.60050
[23] Mao, X.; Sabais, S.; Renshaw, E., Asymptotic behavior of stochastic lotka – volterra model, J. math. anal. appl., 287, 141-156, (2003)
[24] May, R.M., Stability and complexity in model ecosystems, (2001), Princeton University Press NJ
[25] Nindjin, A.F.; Aziz-Alaoui, M.A.; Cadivel, M., Analysis of a predator – prey model with modified leslie – gower and Holling-type II schemes with time delay, Nonlinear anal. real world appl., 7, 1104-1118, (2006) · Zbl 1104.92065
[26] Pielou, E.C., An introduction to mathematical ecology, (1969), Wiley New York · Zbl 0259.92001
[27] Rudnicki, R., Long-time behaviour of a stochastic prey – predator model, Stochastic process. appl., 108, 93-107, (2003) · Zbl 1075.60539
[28] Rudnicki, R.; Pichor, K., Influence of stochastic perturbation on prey – predator systems, Math. biosci., 206, 108-119, (2007) · Zbl 1124.92055
[29] Sun, X.; Wang, Y., Stability analysis of a stochastic logistic model with nonlinear diffusion term, Appl. math. model., 32, 2067-2075, (2008) · Zbl 1145.34348
[30] Takeuchi, Y.; Dub, N.H.; Hieu, N.T.; Sato, K., Evolution of predator – prey systems described by a lotka – volterra equation under random environment, J. math. anal. appl., 323, 938-957, (2006) · Zbl 1113.34042
[31] Upadhyay, R.K.; Rai, V., Crisis-limited chaotic dynamics in ecological systems, Chaos solitons fractals, 12, 205-218, (2001) · Zbl 0977.92033
[32] Upadhyay, R.K.; Iyengar, S.R.K., Effect of seasonality on the dynamics of 2 and 3 species prey – predator system, Nonlinear anal. real world appl., 6, 509-530, (2005) · Zbl 1072.92058
[33] Zhang, S.; Chen, L., A Holling II functional response food chain model with impulsive perturbations, Chaos solitons fractals, 24, 1269-1278, (2005) · Zbl 1086.34043
[34] Zhang, S.; Tan, D.; Chen, L., Chaos in periodically forced Holling type II predator – prey system with impulsive perturbations, Chaos solitons fractals, 28, 367-376, (2006) · Zbl 1083.37537
[35] Tornatore, E.; Manca, L.; Fujita Yashima, H., Asymptotic behavior of the solution of the system of stochastic equations for two competing species, Istit. lombardo accad. sci. lett. rend. A, 136-137, (2002/2003), (2004) 151-183
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.