# zbMATH — the first resource for mathematics

Qualitative analysis of a stochastic ratio-dependent predator-prey system. (English) Zbl 1229.92076
This paper studies the stochastic predator-prey population model \begin{aligned} dx(t) &= x(t)\Biggl(a- bx(t)- {cy(t)\over my(t)+ x(t)}\Biggr)\, dt+\alpha x(t)\,dB_1(t),\quad x(0)= x_0> 0,\\ dy(t) &= y(t)\Biggl(- d+{fx(t)\over my(t)+ x(t)}\Biggr)\,dt-\beta y(t)\,dB_2(t),\quad y(0)= y_0> 0,\end{aligned} where $$B_1$$ and $$B_2$$ are independent Brownian motions, $$a$$, $$b$$, $$c$$, $$d$$, $$f$$, $$m$$, $$\alpha$$, $$\beta$$ are positive constants, and $$x(t)$$, $$y(t)$$ represent the populations of prey and predators, respectively. It is proved that the system has a unique positive solution whose mean is uniformly bounded. If $$A\equiv a-{\alpha^2\over 2}-{c\over m}> 0$$ and $$B\equiv f-d-{\beta^2\over 2}> 0$$ , then $\liminf_{t\to\infty}\;t^{-1}\int^t_0 y(s)/x(s)\,ds$ is positive and $$\lim_{t\to\infty} t^{-1}\int^t_0 x(s)\,ds$$ is finite and positive a.s.; if $$A< 0$$, then $$\lim_{t\to\infty} x(t)= 0$$ and $$\lim_{t\to\infty} y(t)= 0$$ a.s.; and if $$A> 0$$ and $$B< 0$$, then $$\lim_{t\to\infty}y(t)= 0$$ and $$\lim_{t\to\infty} t^{-1} \int^t_0 x(s)\,ds$$ is finite and positive a.s. Results of numerical simulations are presented to show that the populations exhibit this behavior.

##### MSC:
 92D40 Ecology 34F05 Ordinary differential equations and systems with randomness 65C20 Probabilistic models, generic numerical methods in probability and statistics
Full Text:
##### References:
 [1] Freedman, H.I., Deterministic mathematical models in population ecology, (1980), Marcel Dekker New York · Zbl 0448.92023 [2] Arditi, R.; Ginzburg, L.R., Coupling in predator – prey dynamics: ratio-dependence, J. theoret. biol., 139, 311-326, (1989) [3] Arditi, R.; Ginzburg, L.R.; Akcakaya, H.R., Variation in plankton densities among lakes: a case for ratio-dependent models, Am. nat., 138, 1287-1296, (1991) [4] Gutierrez, A.P., The physiological basis of ratio-dependent predator – prey theory: a metabolic pool model of nicholson’s blowflies as an example, Ecology, 73, 1552-1563, (1992) [5] Arditi, R.; Perrin, N.; Saiah, H., Functional response and heterogeneities: an experiment test with cladocerans, Oikos, 60, 69-75, (1991) [6] Arditi, R.; Saiah, H., Empirical evidence of the role of heterogeneity in ratio-dependent consumption, Ecology, 73, 1544-1551, (1992) [7] Hanski, I., The functional response of predator: worries about scale, Tree, 6, 141-142, (1991) [8] Beretta, E.; Kuang, Y., Global analysis in some delayed ratio-dependent predator – prey systems, Nonlinear anal., 32, 381-408, (1998) · Zbl 0946.34061 [9] Berezovskaya, F.; Karev, G.; Arditi, R., Parametric analysis of the ratio-dependent predator – prey model, J. math. biol., 43, 221-246, (2001) · Zbl 0995.92043 [10] Fan, M.; Wang, K., Periodicity in a delayed ratio-dependent predator – prey system, J. math. anal. appl., 262, 179-190, (2001) · Zbl 0994.34058 [11] Hsu, S.B.; Hwang, T.W.; Kuang, Y., Global analysis of the michaelis – menten ratio-dependent predator – prey system, J. math. biol., 42, 489-506, (2001) · Zbl 0984.92035 [12] Hsu, S.B.; Hwang, T.W.; Kuang, Y., Rich dynamics of a ratio-dependent one prey two predator model, J. math. biol., 43, 377-396, (2001) · Zbl 1007.34054 [13] Kuang, Y.; Beretta, E., Global qualitative analysis of a ratio-dependent predator – prey system, J. math. biol., 36, 389-406, (1998) · Zbl 0895.92032 [14] Xu, R.; Chen, L., Persistence and stability for a two-species ratio-dependent predator – prey system with time delay in a two-patch environment, Comput. math. appl., 40, 577-588, (2000) · Zbl 0949.92028 [15] Xu, R.; Chen, L., Persistence and global stability for $$n$$-species ratio-dependent predator – prey system with time delays, J. math. anal. appl., 275, 27-43, (2002) · Zbl 1039.34069 [16] Carletti, M.; Burrage, K.; Burrage, P.M., Numerical simulation of stochastic ordinary differential equations in biomathematical modelling, Math. comput. simulation, 64, 271-277, (2004) · Zbl 1039.65005 [17] Renshaw, E., Modelling biological populations in space and time, (1991), Cambridge University Press Cambridge · Zbl 0754.92018 [18] Beretta, E.; Carletti, M.; Solimano, F., On the effects of environmental fluctuations in a simple model of bacteria – bacteriophage interaction, Can. appl. math. Q., 8, 321-366, (2000) · Zbl 1049.34521 [19] Carletti, M., Numerical simulation of a Campbell-like stochastic delay model for bacteriophage infection, Math. med. biol., 23, 297-310, (2006) · Zbl 1117.92032 [20] Mao, X.; Marion, G.; Renshaw, E., Environmental Brownian noise suppresses explosions in population dynamics, Stochastic process. appl., 97, 95-110, (2002) · Zbl 1058.60046 [21] Bandyopadhyay, M.; Chattopadhyay, J., Ratio-dependent predator – prey model: effect of environmental fluctuation and stability, Nonlinearity, 18, 913-936, (2005) · Zbl 1078.34035 [22] Tapan, S.; Malay, B., Dynamical analysis of a delayed ratio-dependent prey – predator model within fluctuating environment, Appl. math. comput., 196, 458-478, (2008) · Zbl 1153.34051 [23] Tapaswi, P.K.; Mukhopadhyay, A., Effects of environmental fluctuation on plankton allelopathy, J. math. biol., 39, 39-58, (1999) · Zbl 0929.92036 [24] Arnold, L., Stochastic differential equations: theory and applications, (1972), Wiley New York [25] Friedman, A., Stochastic differential equations and their applications, (1976), Academic Press New York [26] Mao, X., Stochastic differential equations and applications, (1997), Horwood New York · Zbl 0874.60050 [27] Ikeda, N., Stochastic differential equations and diffusion processes, (1981), North-Holland Amsterdam [28] Jiang, D.; Shi, N.; Li, X., Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation, J. math. anal. appl., 340, 588-597, (2008) · Zbl 1140.60032 [29] Chen, L.; Chen, J., Nonlinear biological dynamical system, (1993), Science Press Beijing [30] Ryszard, R., Long-time behaviour of a stochastic prey – predator model, Stochastic process. appl., 108, 93-107, (2003) · Zbl 1075.60539 [31] Higham, D.J., An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM rev., 43, 525-546, (2001) · Zbl 0979.65007 [32] Ryszard, R.; Katarzyna, P., Influence of stochastic perturbation on prey – predator systems, Math. biosci., 206, 108-119, (2007) · Zbl 1124.92055 [33] Klebaner, F.C., Introduction to stochastic calculus with applications, (1998), Imperial College Press London · Zbl 0926.60002
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.