About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 738342, 10 pages
http://dx.doi.org/10.1155/2013/738342
Research Article

Stochastic Analysis of a Hassell-Varley Type Predation Model

College of Sciences, Nanjing University of Technology, Nanjing 211816, China

Received 14 November 2013; Accepted 3 December 2013

Academic Editor: Weiming Wang

Copyright © 2013 Feng Rao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We investigate a Hassell-Varley type predator-prey model with stochastic perturbations. By perturbing the growth rate of prey population and death rate of predator population with white noise terms, we construct a stochastic differential equation model to discuss the effects of the environmental noise on the dynamical behaviors. Applying the comparison theorem of stochastic equations and Itô’s formula, the unique positive global solution to the model for any positive initial value is obtained. We find out some sufficient conditions for stochastically asymptotically boundedness, permanence, persistence in mean and extinction of the solution. Furthermore, a series of numerical simulations to illustrate our mathematical findings are presented. The results indicate that the stochastic perturbations do not cause drastic changes of the dynamics in the deterministic model when the noise intensity is small under some conditions, but while the noise intensity is sufficiently large, the species may die out, which does not happen in the deterministic model.

1. Introduction

It is well known that predator-prey interaction is one of basic interspecies relations for ecosystems, and it is also the basic block of more complicated food chain, food web, and biophysical network structure [1]. Because of the universal existence of predator and prey and their importance in ecology, the dynamical relationship between them has long been and will continue to be one of the dominant themes [2, 3].

The classical predator-prey model has received extensive attentions from mathematicians as well as ecologists [47], and it can be expressed by a model of nonlinear ordinary differential equations as follows: where and denote the density of prey and predator population at time , respectively. Parameters , , and are positive constants. stands for capturing rate of prey by predator, is conversion rate of prey into predator, and is the natural death rate of the predator. The function represents the density-dependent specific growth rate of prey in absence of predator. The amount of prey biomass consumed by each predator per unit of time is described by the functional response .

In this paper, we consider the usual logistic form of the growth function for prey in the absence of predator as where is the natural growth rate of prey and is the environmental carrying capacity. The functional response is taken as which is called the Hassell-Varley type functional response and is the Hassell-Varley constant [8] and stands for half capturing saturation constant. The predator-prey model with Hassell-Varley type functional response has been studied in the ecological literature [6, 911].

For more biological motivation in population dynamics, we take into account the density-dependence of predator population. And the corresponding Hassell-Varley type predator-prey model is described by the following form: where stands for the density-dependence of the predator population and .

On the other hand, most natural phenomena do not follow strictly deterministic laws but rather oscillate randomly about some average. So that the population density never attains a fixed value with the advancement of time but rather exhibits continuous oscillation around some average values [12, 13]. In fact, there are many benefits to be gained by using stochastic models because real life is full of random fluctuations (i.e., the effects of noise), which undeniably arise from either environmental variability or internal species. The basic mechanism and factors of population growth like resources and vital rates—birth, death, immigration, and emigration—change nondeterministically due to continuous fluctuations in the environment (e.g., variation in intensity of sunlight, temperature, water level, etc.) [2, 3, 14]. Recent advances in stochastic differential equations enable a lot of authors to introduce noise into the model of physical phenomena, whether it is a random noise in the system of differential equations or environmental fluctuations in parameters [1531]. So far as our knowledge is concerned, the work of a modified Hassell-Varley type predator-prey model with stochastic perturbations seems rare. Motivated by these, we attempt to study the stochastic behaviors of the modified Hassell-Varley type predation model in a random fluctuating environment.

The organization of this paper is as follows. In Section 2, we present a stochastic model corresponding to the deterministic model (4) and discuss it in detail. In Section 3, we use numerical simulations to reveal the influence of noise on the dynamical behaviors of the model. A brief discussion is given in Section 4.

2. The Stochastic Model and Analysis

In this section, we investigate the effects of fluctuating environments on the dynamical behaviors of model (4). Assuming that random fluctuations in the environment would display themselves as fluctuations in the growth rate of prey population and in the death rate of predator population , then the parameters and in model (4) can be replaced by In this way, model (4) will be reduced to the following form: where and are known as the intensities of environmental noise and is a standard white noise; that is, is a Brownian motion defined in a complete probability space with a filtration satisfying the usual conditions (i.e., it is right continuous and increasing while contains all -null sets) [14].

2.1. Positive and Global Solution

For model (6), there is a positive local solution.

Lemma 1. There is a unique local solution for to model (6) almost surely for initial value , where is the explosion time.

The proof of this lemma is rather standard and hence is omitted.

Lemma 1 only tells us that there is a unique positive local solution to model (6). Next, we show that this solution is global which is more interesting.

In particular, let us consider the one-dimensional stochastic population model there is an explicit solution From model (6), we have By the comparison theorem of stochastic equations [14], we have a.s. .

Besides, for the following equation there is a unique solution as In model (6), for , we can get then a.s. .

Consequently, we obtain

On the other hand, the equation has a unique solution as follows: Considering the predator population in model (6), we have By the comparison theorem, we obtain a.s. ; then

From the representation of solutions , , , and , we can see that they are all existence for ; that is, . Therefore, we have the following theorem to show that the positive solution of model (6) is global, which is essential for a population system.

Theorem 2. There is a unique positive solution of model (6) almost surely for any initial value . Moreover there exist functions , , , and defined as (8), (11), (15), and (17) such that

2.2. Stochastic Boundedness

In this subsection, we show that the solution of model (6) with any positive initial value is uniformly bounded in mean.

Theorem 3. The solution of model (6) with any positive initial value has the property that

Proof. From (7), we obtain combining , then
Set then Integrating the above equation from to , we obtain and taking expectations leads to then By the comparison theorem, we can get Therefore, we obtain This completes the proof.

2.3. The Long Time Behavior

It is well known that the property of permanence is more desirable since it means the long time survival in a population dynamics. Now, the definition of stochastic permanence will be given below [32, 33].

Definition 4. The solution of model (6) is said to be stochastically permanent, if, for any , there exists a pair of positive constants and such that, for any initial value , the solution to model (6) has the properties that

Lemma 5. For any initial value , the solution satisfies that where is a positive constant and are arbitrary positive constants satisfying

Proof. Set a function for ; using Itô’s formula, we have Choosing a positive constant and by Itô’s formula, we get
Let be sufficiently small such that it satisfies (31); by Itô’s formula, then Based on the following inequality, Therefore, we obtain There exists a positive constant such that ; then So, we can get
In addition, we know that ; consequently, The proof is complete.

Based on the results of Theorem 3, Lemma 5, and the Chebyshev inequality [14], we can obtain the following theorem.

Theorem 6. Assume that ; the solution of model (6) is stochastically permanent.

In a view of ecology, the coexistence of species may be a good situation. In the following, we consider the stochastic persistence (i.e., stochastic persistence in mean) of the species.

Theorem 7. Assume that holds, for any initial value ; then the solution to model (6) has the property

Proof. Denoting and by Itô’s formula, we obtain Then, we have And we get Dividing on both sides of the previously mentioned inequality yields
Letting , we know that Then we obtain
On the other hand, dividing on both sides and letting , then From the above results, inequality (41) holds.

Theorem 8. Assume that holds and that is the solution of model (6) for any initial value ; then which are stable in time average.

Proof. Denoting and by Itô’s formula, we have Then, Dividing on both sides yields we obtain
Furthermore, then we can get Dividing on both sides, we obtain

2.4. Extinction

From (17), if , then a.s. Moreover, from Theorem 7 and (49), we know that if holds, then which implies that there are a and a positive constant such that a.s. for . Besides, for all , there are and such that and for . Then we obtain which implies that

From Theorem 7 and (47), we have Therefore, by the arbitrary of , we get

Combining the above arguments, we can get the theorem as follows.

Theorem 9. Let be the solution of model (6) with any initial value . If and , then

Furthermore, set and ; for the first equation of model (6) we have Taking the comparison theorem of stochastic equations and the theory of diffusion processes [14], then a.s.; that is,

Similarly, we obtain If not, then there is a positive constant such that Hence, for any given , there exist and a set such that and for . Therefore, By the same reasoning as previously stated, we can get a.s.; that is, There is a contradiction; hence (66) is true.

Based on the above, we obtain the following theorem which means that if the noise satisfies some conditions, then both species and of model (6) will die out.

Theorem 10. Let be the solution of model (6) with any initial value . If , then

3. Numerical Simulations

In this section, we perform some numerical simulations for model (6) with environmental noise to illustrate the previously mentioned analytical findings by referring to the method mentioned in Higham [34]. Next, we consider the discretization equations where and are the Gaussian random variables .

When choosing the values of parameters , , , , , , , and for model (4), which has three equilibria in the positive quadrant, where (total extinct) and (extinct of the predator or prey only) are saddle points, (coexistence of the prey and predator) is globally asymptotically stable. The trajectory of the prey and predator population of model (4) is shown in Figure 1.

738342.fig.001
Figure 1: Phase portrait of model (4). Other parameters are taken as , , , , , , , and . The horizontal axis is prey population and the vertical axis is predator population . and are two saddle points; is stable.

Figure 2 shows the time-series plots of model (6) with different noise intensities and . When choosing (Figure 2(a)) and , (Figure 2(b)), from Theorem 6, we know that the positive solution of model (6) is stochastically permanent, which means that stochastic perturbations do not change the permanence of the deterministic model (4). Moreover, from Theorem 7, the model will be stochastically persistent in mean. In other words, we can use the deterministic model (4) describing the dynamics of the stochastic model when the noise intensities and are small. From Figure 2(b), we can observe that the violent fluctuations appear as the noise intensities further increase. It means that noise has strong destabilizing effects on the model and the amplitude of the fluctuations in population density of prey and predator species increases obviously, implying instability of the coexisting equilibrium point in the stochastic environment.

fig2
Figure 2: Solutions of model (6) with different noise and other parameters are the same as those of Figure 1 and initial condition . (a) and (b) , .

In Figure 3, we continue to choose different noise intensities and to consider the effects of noise to model (6). When choosing and (Figure 3(a)), the conditions of Theorem 9 are satisfied; then we can find that prey population of model (6) is permanent and predator population will die out. Choosing and in Figure 3(b), which satisfies the conditions of Theorem 10, both species and in model (6) become extinct. That is to say, big noise can make the two species die out. From the above numerical results and by Theorems 3, 6, 7, 9, and 10, we conclude that for some noise intensities and the dynamical behaviors of stochastically ultimately boundedness, permanence, persistence in mean, and extinction can be observed in model (6).

fig3
Figure 3: Solutions of model (6) with different noise and other parameters are the same as those of Figure 1. (a) , and (b) , .

4. Conclusions and Remarks

In this paper, we consider a stochastic Hassell-Varley type predator-prey model. The value of this study lies in twofolds. First, it verifies some relevant properties of the corresponding stochastic model (6), which shows the global existence, boundedness and stochastic permanence, persistence in mean, and extinction of the positive solution. Second, it illustrates the dynamics of the model via numerical simulations, which shows that if the noise is not large and satisfies some conditions, the stochastic perturbations do not cause drastic changes of the dynamics in the deterministic model (4), while if the noise is sufficiently large and satisfies some conditions, it will force two species in the model to die out.

In order to study the stochastic model (6), we perturb the deterministic model (4) by incorporating white noise terms in the growth rate of prey population and in the death rate of predator population. Establishing a Lyapunov function, there is a unique positive solution to the model for any positive initial value. Applying Itô’s formula, we derive that, under some conditions, the solution of model (6) is stochastically bounded, permanent, and stochastic persistent in mean and extinct. For the fixed parameters , , , , , , , and , some conditions depend on the intensities of noise and noise . When the noise intensities satisfy some conditions of Theorem 9, we can find that prey population of model (6) is permanent and predator population will die out (see Figure 3(a)), while with the noise intensities increasing which satisfy the conditions of Theorem 10, from Figure 3(b), two species and will die out. In other words, when the noise satisfies some conditions of Theorems 6 and 7 and is not sufficiently large, the populations and may be stochastically permanent and persistent in mean, while large noise satisfying the conditions of Theorems 9 and 10 will force the population to become extinct. Our complete analysis of the stochastic model will give some suggestions to the studies of the population dynamics.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank the editor and the anonymous referees for the very helpful suggestions and comments which led to improvement of their original paper. This research was supported by the National Science Foundation of China (no. 11301263).

References

  1. A. A. Berryman, “The origins and evolution of predator-prey theory,” Ecology, vol. 73, no. 5, pp. 1530–1535, 1992. View at Scopus
  2. R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, NJ, USA, 2001.
  3. J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, vol. 18, Springer, New York, NY, USA, 3rd edition, 2003. View at MathSciNet
  4. E. Beretta and Y. Kuang, “Global analyses in some delayed ratio-dependent predator-prey systems,” Nonlinear Analysis. Theory, Methods & Applications, vol. 32, no. 3, pp. 381–408, 1998. View at Publisher · View at Google Scholar · View at MathSciNet
  5. K. Mischaikow and G. Wolkowicz, “A predator-prey system involving group defense: a connection matrix approach,” Nonlinear Analysis. Theory, Methods & Applications, vol. 14, no. 11, pp. 955–969, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  6. C. Cosner, D. L. Deangelis, J. S. Ault, and D. B. Olson, “Effects of spatial grouping on the functional response of predators,” Theoretical Population Biology, vol. 56, no. 1, pp. 65–75, 1999. View at Publisher · View at Google Scholar · View at Scopus
  7. M. Bandyopadhyay and J. Chattopadhyay, “Ratio-dependent predator-prey model: effect of environmental fluctuation and stability,” Nonlinearity, vol. 18, no. 2, pp. 913–936, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. M. P. Hassell and G. C. Varley, “New inductive population model for insect parasites and its bearing on biological control,” Nature, vol. 223, no. 5211, pp. 1133–1137, 1969. View at Publisher · View at Google Scholar · View at Scopus
  9. P. A. Abrams and L. R. Ginzburg, “The nature of predation: prey dependent, ratio dependent or neither?” Trends in Ecology and Evolution, vol. 15, no. 8, pp. 337–341, 2000. View at Publisher · View at Google Scholar · View at Scopus
  10. G. T. Skalski and J. F. Gilliam, “Functional responses with predator interference: viable alternatives to the Holling type II model,” Ecology, vol. 82, no. 11, pp. 3083–3092, 2001. View at Scopus
  11. W. J. Sutherland, “Aggregation and the “ideal free” distribution,” The Journal of Animal Ecology, vol. 52, no. 3, pp. 821–828, 1983. View at Scopus
  12. T. C. Gard, “Persistence in stochastic food web models,” Bulletin of Mathematical Biology, vol. 46, no. 3, pp. 357–370, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. T. C. Gard, “Stability for multispecies population models in random environments,” Nonlinear Analysis. Theory, Methods & Applications, vol. 10, no. 12, pp. 1411–1419, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. B. K. Øksendal, Stochastic Differential Equations: An Introduction with Applications, Springer, New York, NY, USA, 4th edition, 1995. View at MathSciNet
  15. K. Liu and X. Mao, “Exponential stability of non-linear stochastic evolution equations,” Stochastic Processes and Their Applications, vol. 78, no. 2, pp. 173–193, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. R. R. Sarkar, “A stochastic model for autotroph-herbivore system with nutrient reclycing,” Ecological Modelling, vol. 178, no. 3-4, pp. 429–440, 2004. View at Publisher · View at Google Scholar · View at Scopus
  17. A. Bahar and X. Mao, “Stochastic delay Lotka-Volterra model,” Journal of Mathematical Analysis and Applications, vol. 292, no. 2, pp. 364–380, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. E. Tornatore, S. M. Buccellato, and P. Vetro, “Stability of a stochastic SIR system,” Physica A, vol. 354, no. 1–4, pp. 111–126, 2005. View at Publisher · View at Google Scholar · View at Scopus
  19. D. Q. Jiang and N. Z. Shi, “A note on nonautonomous logistic equation with random perturbation,” Journal of Mathematical Analysis and Applications, vol. 303, no. 1, pp. 164–172, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. S. A. L. M. Kooijman, J. Grasman, and B. W. Kooi, “A new class of non-linear stochastic population models with mass conservation,” Mathematical Biosciences, vol. 210, no. 2, pp. 378–394, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  21. N. Dalal, D. Greenhalgh, and X. R. Mao, “A stochastic model for internal HIV dynamics,” Journal of Mathematical Analysis and Applications, vol. 341, no. 2, pp. 1084–1101, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. S. L. Pang, F. Q. Deng, and X. R. Mao, “Asymptotic properties of stochastic population dynamics,” Dynamics of Continuous, Discrete & Impulsive Systems A, vol. 15, no. 5, pp. 603–620, 2008. View at Zentralblatt MATH · View at MathSciNet
  23. F. Rao, W. M. Wang, and Z. Q. Li, “Spatiotemporal complexity of a predator-prey system with the effect of noise and external forcing,” Chaos, Solitons & Fractals, vol. 41, no. 4, pp. 1634–1644, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. X.-Z. Meng, “Stability of a novel stochastic epidemic model with double epidemic hypothesis,” Applied Mathematics and Computation, vol. 217, no. 2, pp. 506–515, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. J. Lv and K. Wang, “Asymptotic properties of a stochastic predator-prey system with Holling II functional response,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 10, pp. 4037–4048, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. C. Y. Ji, D. Q. Jiang, and X. Y. Li, “Qualitative analysis of a stochastic ratio-dependent predator-prey system,” Journal of Computational and Applied Mathematics, vol. 235, no. 5, pp. 1326–1341, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. X. X. Wang, H. L. Huang, Y. L. Cai, and W. M. Wang, “The complex dynamics of a stochastic predatorprey model,” Abstract and Applied Analysis, vol. 2012, Article ID 401031, 24 pages, 2012. View at Publisher · View at Google Scholar
  28. M. Liu and K. Wang, “Dynamics of a Leslie-Gower Holling-type II predator-prey system with Lévy jumps,” Nonlinear Analysis. Theory, Methods & Applications, vol. 85, pp. 204–213, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  29. D. Jiang, C. Ji, X. Li, and D. O'Regan, “Analysis of autonomous Lotka-Volterra competition systems with random perturbation,” Journal of Mathematical Analysis and Applications, vol. 390, no. 2, pp. 582–595, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. M. Liu and K. Wang, “Analysis of a stochastic autonomous mutualism model,” Journal of Mathematical Analysis and Applications, vol. 402, no. 1, pp. 392–403, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. F. Rao, “Dynamical analysis of a stochastic predator-prey model with an Allee effect,” Abstract and Applied Analysis, vol. 2013, Article ID 340980, 10 pages, 2013. View at Publisher · View at Google Scholar
  32. D. Q. Jiang, N. Z. Shi, and X. Y. Li, “Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation,” Journal of Mathematical Analysis and Applications, vol. 340, no. 1, pp. 588–597, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. X. Y. Li and X. R. Mao, “Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation,” Discrete and Continuous Dynamical Systems A, vol. 24, no. 2, pp. 523–545, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  34. D. J. Higham, “An algorithmic introduction to numerical simulation of stochastic differential equations,” SIAM Review, vol. 43, no. 3, pp. 525–546, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet