Influence of spatiotemporal and noise on dynamics of a two species commensalism model with optimal harvesting

 

G Basava Kumar, M N Srinivas

School of Advanced Sciences, VIT University, Vellore, Tamil Nadu, India

 

ABSTRACT:

The focus of study is on the dynamics of interaction between commensal and host species in an aquatic ecological system with harvesting, diffusion and stochastic attributes. All possible equilibrium points of the system are identified. The stability both local and global of the commensal-host system has been examined around the interior equilibrium point. The possibility of existence of bio-economic equilibrium is discussed. The optimal harvesting levels are identified. The role of diffusion parameters on stability is analysed. We provided the analytical estimates of the population intensities of fluctuations of the system with Gaussian white noise perturbations by applying Fourier transform technique. Some numerical simulations are also carried out to make lawful the analysis.

 

INTRODUCTION:

Ecology is the study of relationships between living organisms and their environment. Investigations in the discipline of theoretical ecology were initiated by Lotka [1] and Volterra [2]. Several researchers contributed to the enlargement of this area of acquaintance has been expansively accounted in the dissertations of Meyer [3], Cushing [4], Paul colinvaux [5], Kapur [6, 7] etc. The biological dealings can be generally classified as predation, competition, commensalism, ammensalism, neutralism etc. The present investigation is devoted to the analytical study of commensalism between two species. Commensalism is an ecological relationship between two species where one species (S₁) derives a benefit from the other (S₂)   which does not get affected by it. S₁ may be referred as the commensal species while

 

S₂ the host. The oceanic ecosystems are mainly focussed in this study to analyse the noise induced dynamics as well as the diffusive dynamics. The Barnacle-Whale interaction is an example of commensalism where the Barnacle is commensal which affix itself on the host Whales. Barnacles ride on Whales to the nutrient-rich water zones, but neither harm nor benefit to the host by the riders. Similarly Sucker fish-Shark display a commensal-host relationship where the sucker fish attach itself underside of the host Shark. The commensal sucker fish is travelling to distant areas of rich food source and is getting protected from its predators due to this attachment with the host Shark which does not get harmed or benefitted by the Sucker fish. The titan trigger fish stays as a host for small fishes where the commensal small fishes are getting better feeding opportunities as the big rocks are moved by the host.

 

The effect of dispersal and spatial heterogeneity is more important in the population dynamics as it plays major role on the stability of the ecosystems. The qualitative theory on diffusive systems was constructed and developed by many researchers [8-15]. Sining zheng [16] analysed a predator-prey-mutualist model with diffusion and revealed that the solution is unique and bounded regardless of diffusive mechanism and types of boundary conditions due to the biological assumptions like (i) the finite carrying capacity of mutualist prey and (ii) the mutualist species benefits the mutualist prey by preventing the predation on prey. Sherratt [17] considered a reaction-diffusion model and exhibited the natural generation of spatiotemporal irregularities leading to spatiotemporal chaos due to selection of unstable wave and also predicted the oscillatory behaviour of reaction-diffusion equations. Dubey and Hussain [18] analysed a competitive model of two species dependent on resource in the industrial environment and shown that the solution of the system converges to steady state more rapidly with the increase in diffusion co-efficients. Dhar [19] investigated a prey-predator model in the two patch habitat with diffusion where predator is partially dependent on prey and guaranteed the existence of positive, monotonic, continuous steady state solution with continuous matching at the interface for both the species separately. Mukhopadhyay et al. [20] studied a delayed-diffusive predator-prey model and ensured the stability of the disease-free steady state due to the guiding factors (i) high diffusivity of infected prey and (ii) low diffusivity of susceptible prey and predator. Dubey and Hussain [21] examined the survival of species dependent on resources in a polluted environment and observed that solution of the system approaches the equilibrium level faster in the case of diffusion than in the case of no diffusion. Yihong Du and Junping Shi [22] considered a predator-prey model with protection zone and observed that if the protection zone size is below than the critical patch size, then the dynamics of the system is similar to the case without protection zone, but it is shown that the size of the prey increases with the size of protection zone. Balram Dubey et al. [23] studied the reaction diffusion mechanism in a prey-predator system and observed that the solution of the system converges to its equilibrium point faster with the two dimensional diffusion than the case of one dimensional diffusion. Shaban Aly et al. [24] examined a ratio-dependent predator-prey model with diffusion and concluded that the kinetic system is stable and there exists Turing type instability at a critical value of diffusion constants. Jia Liu and Zhigui Lin [25] studied a prey-predator model with Holling type-III functional response and cross diffusion subjected to Neumann boundary conditions and highlighted the existence and non-existence of non-constant positive steady state as the co-efficient of cross diffusion are varied. Nurul Huda Gazi and Kalyan Das [26] considered an autotroph-herbivore system with delay and diffusion and analysed the asymptotic stability and recognized the switching to instability with bifurcation where the diffusion controls the delay parameter directly.

Holling [27] highlighted the effect of noise in ecological dynamics and flexibility. The noise may arise from stochastic disturbances on the external environment, and through the influence of noise, the species dynamics is stochastic apparently. In 1996, Ripa et al. [28] inspected the noise color and the risk of population extinction in a prey–predator system widely. In 2003 Xu et al.[29] investigated a three species food chain system with a white variance spectrum which contains no temporal auto correlation and is fundamentally a series of independent random numbers about the population dynamics and the color environmental noise. Wang et al. [30] found the effect of colored noise on spatiotemporal dynamics of biological invasion in a predator–prey system.

 

The present work is motivated by [30-31] to study the effect of ecological fluctuations on the commensal-host aquatic ecosystem and the paper is organized as follows: In Section 2, the mathematical model of commensalism system with noise and harvesting is formulated. The steady states and stability (both local and global) have been discussed in section 3. The bionomic equilibrium is obtained in section 4. The optimal control strategy on harvesting is discussed in section 5. The stability under stochastic environment has been studied in section 6. Section 7 consists of diffusion analysis. The conclusions are provided in section 8.

 

8. CONCLUSION:

The commensal-host model is considered with harvesting, diffusion and stochastic effects. All possible equilibrium points of the system are found and the stability analysis is carried out. The interior equilibrium is locally asymptotically stable if it exists. It is exhibited in the   figures 1 and 2. The parametric region of global stability of the positive equilibrium point is identified. The bionomic equilibrium is also obtained which will indicate the maximal limits of harvesting. The harvesting levels which optimize the profit and population growth are found. The population variances are computed which describes the system stability under random environmental fluctuations. The numerical simulations in figures 3 and 4 are exhibiting the periodic fluctuations in population around the positive equilibrium point in the presence of noise. It can also be observed that if the amplitude of noise is more, then the population densities fluctuating with high intensity and if the strength of the noise is less, then that results in low intensity fluctuations.

                             

9. REFERENCES:

1.     Lotka AJ. Elements of Physical Biology. Williams and Wilking Company; 1925.

2.     Volterra V. Leconssen La Theorie Mathematique De La Leitte Pou Lavie. Paris: Gouthier Villars; 1931.

3.     Meyer WJ. Concepts of Mathematical Modeling. Mc.Graw-Hill; 1985.

4.     Cushing JM. Integro-Differential Equations and Delay Models in Population Dynamics,  Lecture Notes in Bio-Mathematics, Vol.20,     Springer Verlag; 1997.

5.     Paul Colinvaux A. Ecology.  New York: John Wiley; 1986.

6.     Kapur JN. Mathematical Modelling in Biology and Medicine. Affiliated East West; 1985.

7.     Levin S.A., Dispersion and population interactions, Am. Nat.. 1974; 108:  207-228.

8.     Levin S.A., Spatial patterning and the structure of ecological communities, Lectures on mathematics in the life sciences.  1976; 8.

9.     Gopalsamy K., Competition, dispersion & co-existence, Math. Bio. Science. 1977; 33: 25-33.

10.  Hastings A., Global Stability in Lotka-Volterra Systems with Diffusion, J. Math. Biol. 1978; 6: 163-168.

11.  Hallam T.G., A temporal study of diffusion effects on population modeled by quadratic growth, Nonl. Anal. Theory Methods & Appl. 1979; 3: 123-133.

12.  Okubo A., Diffusion and Ecological Problem: Mathematical models, Springer-Verlag, New York, 1980.

13.  Pao C.V., Coexistence and stability of a competition diffusion system in population dynamics, J. Math. Anal. Appl. 1981; 83: 54-76.

14.  Shukla V.P., Shukla J.B., Multispecies food webs with Diffusion, J. Math, Biol. 1982; 13: 339-344.

15.  Rothe F., Global solution of Reaction Diffusion systems, Lecture Notes in Math. 1072,  Springer-Verlag, Berlin Heidelberg, New York, Tokyo, 1984.

16.  Sining Zheng, A reaction diffusion system of a predator-prey-mutualist model, Math. Biosciences. 1986; 78:  217-245.

17.  Sherratt J.A., Periodic waves in reaction-diffusion on models of oscillatory biological systems. 1996; 11:  61-80.

18.  Dubey B., Hussain J., Nonlinear models for the survival of two competing species dependent on resource in industrial environments, Nonl.Anal.: Real world Appl. 2003; 4: 21-44.

19.  Dhar J., A prey-predator model with diffusion and supplementary resource for the prey in a two patch environment, Math. Model. And Anal. 2004; 9 (1) : 9-24.

20.  Mukhopadhyay B., Bhattacharya R., Dynamics of a delay-diffusion prey-predator model with disease in the prey, J. Appl. Math. & Computing. 2005; 17 (1-2) : 361-377.

21.  Dubey B., Hussain J., Modelling the survival of species dependent on a resource in a polluted environment, Nonlinear Analysis: Real World Applications. 2006; 7 : 187-210.

22.  Yihong Du, Junping Shi, A diffusive predator- prey model with a protection zone, J. Differential Equations.  2006; 229: 63-91.

23.  Balram Dubey, Nitu Kumari, Ranjit Kumar Upadhyay, Spatiotemporal pattern formation in a diffusive predator-prey system: an analytical approach, J. Appl. Math. Comput. 2009; 31: 413-432.

24.  Shaban Aly, Imbunm Kim, Dangwoo Sheen, Turing Instability for an ratio-dependent predator-prey model with diffusion, as a preprint submission to Elsevier in 2009.

25.  Jia Liu, Zhigui Lin, Stationary patterns for a predator-prey model with Holling type-III response function and cross-diffusion, Bull. Korean Math. Soc. 2010; 47(2) : 251-261.

26.  Nurul Huda Gazi, Kalyan Das, Control of parameters of a delayed diffusive autotroph herbivore system, Journal of Biological systems. 2010; 18(2) : 509-529.

27.  C.S. Holling, “Resilience and stability of ecological systems”,  Ann. Rev. Ecol. Syst.  1973; 4: 1–23.

28.  J. Ripa, P. Lundberg, “Noise colour and the risk of population extinction”,  Proc. R. Soc. Lond. B. 1996; 263: 1751–1753.

29.  C. Xu, Z. Li, “Population dynamics and the color of environmental noise: a study on a three-species food chain system”, Ecol. Res. 2003; 18: 145–154.

30.  Wenting Wang, Wenlong Li, Zizhen Li, Hui Zhang, “The effect of colored noise on spatiotemporal dynamics of biological invasion in a diffusive predator–prey system”, Bio Syst. 2011; 104: 48–56.                                                                                                           

 

31.  Luca Ridolf, Paolo, Paolo D’Odorico, Francesco Laio, “Noise induced phenomena in the environmental sciences”,  Cambridge University Press, 2011.

32.  Kar TK, Ghosh B. Sustainability and optimal control of an exploited prey-predator system through provision of alternative food to predator.  Biosystems. 2012; 109(2): 220-232.

33.  Kar TK, Ghosh B. Impacts of maximum sustainable yield policy to prey-predator systems. Ecological modeling. 2013; 250:  134-142.

34.  Ghosh B, Kar TK. Possible ecosystems impacts of applying maximum sustainable yield policy in food chain models. Journal of theoretical biology. 2013; 329:  6-14.

35.  Ghosh B, Kar TK. Maximum sustainable yield and species extinction in a prey predator system: some new results. Journal of biological physics 2013; 39(3):453-467.

36.  Balram Dubey, Nitu Kumari, Ranjit Kumar Upadhyay, Spatiotemporal pattern formation in a diffusive predator-prey system: an analytical approach, J. Appl. Math. Comput. 2009; 31:  413-432.

37.  Nisbet RM, Gurney WSC. Modelling Fluctuating Populations. New York :John Wiley; 1982.

38.  Carletti M. Numerical solution of stochastic differential problems in the biosciences. J Comput Appl Math. 2006; 185: 422–440.

 

 

Accepted on 07.10.2016           © RJPT All right reserved

Research J. Pharm. and Tech 2016; 9(10):1717-1726.