# American Institute of Mathematical Sciences

June  2017, 22(4): 1493-1508. doi: 10.3934/dcdsb.2017071

## Optimal harvesting of a stochastic delay competitive model

 1 School of Mathematical Science, Huaiyin Normal University, Huaian 223300, China 2 School of Mathematics and Statistics, Northeast Normal University, Jilin 130024, China

Received  November 2014 Revised  April 2016 Published  February 2017

Fund Project: This research is supported by National Natural Science Foundation of China (Nos. 11301207, 11571136, and 11171081), Natural Science Foundation of Jiangsu Province (Nos. BK2011407 and BK20130411), Qing Lan Project of Jiangsu Province (2014), Project Funded by China Post-doctoral Science Foundation (2015M571349, 2016T90236), Jiangsu Province “333 High-Level Personnel Training Project”, Science and Technology Support Plan Project of Huaian (HAR2015013).

In this paper an $n$-species stochastic delay competitive model with harvesting is proposed. Some dynamical properties of the model are considered. We first establish sufficient conditions for persistence in the mean of the species. Then asymptotic stability in distribution of the harvesting model is studied. Next the optimal harvesting effort and the maximum harvesting yield are given by using the ergodic approach. Finally the analytical results are illustrated through simulation figures using MATLAB followed by discussions and conclusions.

Citation: Meng Liu, Chuanzhi Bai. Optimal harvesting of a stochastic delay competitive model. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1493-1508. doi: 10.3934/dcdsb.2017071
##### References:
 [1] L. H. R. Alvarez and L. A. Shepp, Optimal harvesting of stochastically fluctuating populations, J. Math. Biol., 37 (1998), 155-177.  doi: 10.1007/s002850050124. [2] A. Bahar and X. Mao, Stochastic delay population dynamics, Int. J. Pure Appl. Math., 11 (2004), 377-400. [3] J. Bao, Z. Hou and C. Yuan, Stability in distribution of neutral stochastic differential delay equations with Markovian switching, Statist. Probab. Lett., 79 (2009), 1663-1673.  doi: 10.1016/j.spl.2009.04.006. [4] J. Bao, X. Mao, G. Yin and C. Y. uan, Competitive Lotka-Volterra population dynamics with jumps, Nonlinear Anal., 74 (2011), 6601-6616.  doi: 10.1016/j.na.2011.06.043. [5] J. Bao and C. Yuan, Comparison theorem for stochastic differential delay equations with jumps, Acta Appl. Math., 116 (2011), 119-132.  doi: 10.1007/s10440-011-9633-7. [6] I. Barbalat, Systems dequations differentielles d'osci d'oscillations nonlineaires, Revue Roumaine de Mathematiques Pures et Appliquees, 4 (1959), 267-270. [7] J. R. Beddington and R. M. May, Harvesting natural populations in a randomly fluctuating environment, Science, 197 (1977), 463-465. [8] D. K. Bhattacharya and S. Begum, Bionomic equilibrium of two species system, Math. Biosci., 135 (1996), 111-127. [9] C. A. Braumann, Variable effort harvesting models in random environments: Generalization to density-dependent noise intensities, Math. Biosci., 177&178 (2002), 229-245.  doi: 10.1016/S0025-5564(01)00110-9. [10] N. Bruti-Liberati and E. Platen, Monte Carlo simulation for stochastic differential equations, Encyclopedia of Quantitative Finance, 10 (2010), 23-37.  doi: 10.1080/14697680902814233. [11] K. S. Chaudhuri and S. Saha Roy, On the combined harvesting of a prey-predator system, J. Biol. Syst., 4 (1996), 376-389. [12] C. W. Clark, Bioeconomic Modelling and Fisheries Management, Wiley, New York, 1985. [13] H. Crauel and M. Gundlach, Stochastic Dynamics, Springer-Verlag, New York, 1999. doi: 10.1007/b97846. [14] T. C. Gard, Stability for multispecies population models in random environments, Nonlinear Anal., 10 (1986), 1411-1419.  doi: 10.1016/0362-546X(86)90111-2. [15] K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic, Dordrecht, 1992. doi: 10.1007/978-94-015-7920-9. [16] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546.  doi: 10.1137/S0036144500378302. [17] D. Jiang, C. Ji, X. Li and D. O'Regan, Analysis of autonomous Lotka-Volterra competition systems with random perturbation, J. Math. Anal. Appl., 390 (2012), 582-595.  doi: 10.1016/j.jmaa.2011.12.049. [18] D. Jiang and N. Shi, A note on nonautonomous logistic equation with random perturbation, J. Math. Anal. Appl., 303 (2005), 164-172.  doi: 10.1016/j.jmaa.2004.08.027. [19] T. K. Kar, Influence of environmental noises on the Gompertz model of two species fishery, Ecological Modelling, 17 (2004), 251-272. [20] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993. [21] R. Lande, S. Engen and B. E. Saeher, Optimal harvesting of fluctuating populations with a risk of extinction, Am. Nat., 145 (1995), 728-745. [22] W. Li and K. Wang, Optimal harvesting policy for general stochastic Logistic population model, J. Math. Anal. Appl., 368 (2010), 420-428.  doi: 10.1016/j.jmaa.2010.04.002. [23] M. Liu and C. Bai, Optimal harvesting of a stochastic logistic model with time delay, J. Nonlinear Sci., 25 (2015), 277-289.  doi: 10.1007/s00332-014-9229-2. [24] M. Liu and C. Bai, Optimal harvesting of a stochastic mutualism model with Lévy jumps, Appl. Math. Comput., 276 (2016), 301-309.  doi: 10.1016/j.amc.2015.11.089. [25] M. Liu and C. Bai, Analysis of a stochastic tri-trophic food-chain model with harvesting, J. Math. Biol., 73 (2016), 597-625.  doi: 10.1007/s00285-016-0970-z. [26] M. Liu and M. Fan, Permanence of stochastic lotka-volterra systems, J. Nonlinear Sci. , (2016). doi: 10.1007/s00332-016-9337-2. [27] M. Liu and M. Fan, Stability in distribution of a three-species stochastic cascade predator-prey system with time delays, IMA J. Appl. Math. , (2016). doi: 10.1093/imamat/hxw057. [28] D. Ludwig and J. M. Varah, Optimal harvesting of a randomly fluctuating resource Ⅱ: Numerical methods and results, SIAM J. Appl. Math., 37 (1979), 185-205.  doi: 10.1137/0137012. [29] E. M. Lungu and B. $\emptyset$ksendal, Optimal harvesting from a population in a stochastic crowded environment, Math. Biosci., 145 (1997), 47-75.  doi: 10.1016/S0025-5564(97)00029-1. [30] Y. Lv, R. Yuan and Y. Pei, Dynamics in two nonsmooth predator-preymodels with threshold harvesting, Nonlinear Dyn., 74 (2013), 107-132.  doi: 10.1007/s11071-013-0952-2. [31] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, 2006. doi: 10.1142/p473. [32] R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, New York, 2001. [33] M. Mesterton-Gibbons, On the optimal policy for combined harvesting of predator and prey, Nat. Resour. Model. , 3 (1988), 303. [34] D. Pal, G. S. Mahaptra and G. P. Samanta, Optimal harvesting of prey-predator system with interval biological parameters: A bioeconomic model, Math. Biosci., 241 (2013), 181-187.  doi: 10.1016/j.mbs.2012.11.007. [35] D. Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge University Press, Cambridge, 1996. [36] D. L. Ragogin and G. Brown, Harvest polices and non-market valuation in a predator prey system, J. Environ. Econ. Manag., 12 (1985), 155-168. [37] D. Ryan and F. Hanson, Optimal harvesting of a logistic population in an environment with stochastic jumps, J. Math. Biol., 24 (1986), 259-277.  doi: 10.1007/BF00275637. [38] Q. S. Song, R. Stockbridge and C. Zhu, On optimal harvesting problems in random environments, SIAM J. Control Optim., 49 (2011), 859-889.  doi: 10.1137/100797333. [39] Y. Zhang and Q. Zhang, Dynamic behavior in a delayed stage-structured population model with stochastic fluctuation and harvesting, Nonlinear Dynam., 66 (2011), 231-245.  doi: 10.1007/s11071-010-9923-z. [40] C. Zhu and G. Yin, On hybrid competitive Lotka-Volterra ecosystems, Nonlinear Anal., 71 (2009), e1370-e1379.  doi: 10.1016/j.na.2009.01.166. [41] X. Zou, W. Li and K. Wang, Ergodic method on optimal harvesting for a stochastic Gompertz-type diffusion process, Appl. Math. Lett., 26 (2013), 170-174.  doi: 10.1016/j.aml.2012.08.006. [42] X. Zou and K. Wang, Optimal harvesting for a Logistic population dynamics driven by a Lévy process, J. Optim. Theory Appl., 161 (2014), 969-979.  doi: 10.1007/s10957-013-0451-0. [43] X. Zou and K. Wang, Optimal harvesting for a stochastic regime-switching logistic diffusion system with jumps, Nonlinear Anal. Hybrid Syst., 13 (2014), 32-44.  doi: 10.1016/j.nahs.2014.01.001.

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##### References:
 [1] L. H. R. Alvarez and L. A. Shepp, Optimal harvesting of stochastically fluctuating populations, J. Math. Biol., 37 (1998), 155-177.  doi: 10.1007/s002850050124. [2] A. Bahar and X. Mao, Stochastic delay population dynamics, Int. J. Pure Appl. Math., 11 (2004), 377-400. [3] J. Bao, Z. Hou and C. Yuan, Stability in distribution of neutral stochastic differential delay equations with Markovian switching, Statist. Probab. Lett., 79 (2009), 1663-1673.  doi: 10.1016/j.spl.2009.04.006. [4] J. Bao, X. Mao, G. Yin and C. Y. uan, Competitive Lotka-Volterra population dynamics with jumps, Nonlinear Anal., 74 (2011), 6601-6616.  doi: 10.1016/j.na.2011.06.043. [5] J. Bao and C. Yuan, Comparison theorem for stochastic differential delay equations with jumps, Acta Appl. Math., 116 (2011), 119-132.  doi: 10.1007/s10440-011-9633-7. [6] I. Barbalat, Systems dequations differentielles d'osci d'oscillations nonlineaires, Revue Roumaine de Mathematiques Pures et Appliquees, 4 (1959), 267-270. [7] J. R. Beddington and R. M. May, Harvesting natural populations in a randomly fluctuating environment, Science, 197 (1977), 463-465. [8] D. K. Bhattacharya and S. Begum, Bionomic equilibrium of two species system, Math. Biosci., 135 (1996), 111-127. [9] C. A. Braumann, Variable effort harvesting models in random environments: Generalization to density-dependent noise intensities, Math. Biosci., 177&178 (2002), 229-245.  doi: 10.1016/S0025-5564(01)00110-9. [10] N. Bruti-Liberati and E. Platen, Monte Carlo simulation for stochastic differential equations, Encyclopedia of Quantitative Finance, 10 (2010), 23-37.  doi: 10.1080/14697680902814233. [11] K. S. Chaudhuri and S. Saha Roy, On the combined harvesting of a prey-predator system, J. Biol. Syst., 4 (1996), 376-389. [12] C. W. Clark, Bioeconomic Modelling and Fisheries Management, Wiley, New York, 1985. [13] H. Crauel and M. Gundlach, Stochastic Dynamics, Springer-Verlag, New York, 1999. doi: 10.1007/b97846. [14] T. C. Gard, Stability for multispecies population models in random environments, Nonlinear Anal., 10 (1986), 1411-1419.  doi: 10.1016/0362-546X(86)90111-2. [15] K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic, Dordrecht, 1992. doi: 10.1007/978-94-015-7920-9. [16] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546.  doi: 10.1137/S0036144500378302. [17] D. Jiang, C. Ji, X. Li and D. O'Regan, Analysis of autonomous Lotka-Volterra competition systems with random perturbation, J. Math. Anal. Appl., 390 (2012), 582-595.  doi: 10.1016/j.jmaa.2011.12.049. [18] D. Jiang and N. Shi, A note on nonautonomous logistic equation with random perturbation, J. Math. Anal. Appl., 303 (2005), 164-172.  doi: 10.1016/j.jmaa.2004.08.027. [19] T. K. Kar, Influence of environmental noises on the Gompertz model of two species fishery, Ecological Modelling, 17 (2004), 251-272. [20] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993. [21] R. Lande, S. Engen and B. E. Saeher, Optimal harvesting of fluctuating populations with a risk of extinction, Am. Nat., 145 (1995), 728-745. [22] W. Li and K. Wang, Optimal harvesting policy for general stochastic Logistic population model, J. Math. Anal. Appl., 368 (2010), 420-428.  doi: 10.1016/j.jmaa.2010.04.002. [23] M. Liu and C. Bai, Optimal harvesting of a stochastic logistic model with time delay, J. Nonlinear Sci., 25 (2015), 277-289.  doi: 10.1007/s00332-014-9229-2. [24] M. Liu and C. Bai, Optimal harvesting of a stochastic mutualism model with Lévy jumps, Appl. Math. Comput., 276 (2016), 301-309.  doi: 10.1016/j.amc.2015.11.089. [25] M. Liu and C. Bai, Analysis of a stochastic tri-trophic food-chain model with harvesting, J. Math. Biol., 73 (2016), 597-625.  doi: 10.1007/s00285-016-0970-z. [26] M. Liu and M. Fan, Permanence of stochastic lotka-volterra systems, J. Nonlinear Sci. , (2016). doi: 10.1007/s00332-016-9337-2. [27] M. Liu and M. Fan, Stability in distribution of a three-species stochastic cascade predator-prey system with time delays, IMA J. Appl. Math. , (2016). doi: 10.1093/imamat/hxw057. [28] D. Ludwig and J. M. Varah, Optimal harvesting of a randomly fluctuating resource Ⅱ: Numerical methods and results, SIAM J. Appl. Math., 37 (1979), 185-205.  doi: 10.1137/0137012. [29] E. M. Lungu and B. $\emptyset$ksendal, Optimal harvesting from a population in a stochastic crowded environment, Math. Biosci., 145 (1997), 47-75.  doi: 10.1016/S0025-5564(97)00029-1. [30] Y. Lv, R. Yuan and Y. Pei, Dynamics in two nonsmooth predator-preymodels with threshold harvesting, Nonlinear Dyn., 74 (2013), 107-132.  doi: 10.1007/s11071-013-0952-2. [31] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, 2006. doi: 10.1142/p473. [32] R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, New York, 2001. [33] M. Mesterton-Gibbons, On the optimal policy for combined harvesting of predator and prey, Nat. Resour. Model. , 3 (1988), 303. [34] D. Pal, G. S. Mahaptra and G. P. Samanta, Optimal harvesting of prey-predator system with interval biological parameters: A bioeconomic model, Math. Biosci., 241 (2013), 181-187.  doi: 10.1016/j.mbs.2012.11.007. [35] D. Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge University Press, Cambridge, 1996. [36] D. L. Ragogin and G. Brown, Harvest polices and non-market valuation in a predator prey system, J. Environ. Econ. Manag., 12 (1985), 155-168. [37] D. Ryan and F. Hanson, Optimal harvesting of a logistic population in an environment with stochastic jumps, J. Math. Biol., 24 (1986), 259-277.  doi: 10.1007/BF00275637. [38] Q. S. Song, R. Stockbridge and C. Zhu, On optimal harvesting problems in random environments, SIAM J. Control Optim., 49 (2011), 859-889.  doi: 10.1137/100797333. [39] Y. Zhang and Q. Zhang, Dynamic behavior in a delayed stage-structured population model with stochastic fluctuation and harvesting, Nonlinear Dynam., 66 (2011), 231-245.  doi: 10.1007/s11071-010-9923-z. [40] C. Zhu and G. Yin, On hybrid competitive Lotka-Volterra ecosystems, Nonlinear Anal., 71 (2009), e1370-e1379.  doi: 10.1016/j.na.2009.01.166. [41] X. Zou, W. Li and K. Wang, Ergodic method on optimal harvesting for a stochastic Gompertz-type diffusion process, Appl. Math. Lett., 26 (2013), 170-174.  doi: 10.1016/j.aml.2012.08.006. [42] X. Zou and K. Wang, Optimal harvesting for a Logistic population dynamics driven by a Lévy process, J. Optim. Theory Appl., 161 (2014), 969-979.  doi: 10.1007/s10957-013-0451-0. [43] X. Zou and K. Wang, Optimal harvesting for a stochastic regime-switching logistic diffusion system with jumps, Nonlinear Anal. Hybrid Syst., 13 (2014), 32-44.  doi: 10.1016/j.nahs.2014.01.001.
Solution of model (26) with parameter values given in Table 1 and initial conditions $x_1(\theta)=0.6+0.1\sin \theta, x_2(\theta)=0.2-0.08\sin \theta$, $-10\leq\theta\leq0.$
Distribution of model (26) with the parameter values given in Table 1 and initial conditions $x_1(\theta)=0.5-0.1\sin \theta, x_2(\theta)=0.2+0.08\sin \theta$, $-10\leq\theta\leq0$. (a) is with $t=100$; (b) is with $t=400$; (c) is with $t=700$; (d) is with $t=1000$.
$E\big[h_1x_1(t)+h_2x_2(t)\big]$ of model (26) with the parameter values given in Table 2 and initial conditions $x_1(\theta)=0.5-0.1\sin \theta, x_2(\theta)=0.2+0.08\sin \theta$, $-10\leq\theta\leq0$. Green line is with $h_1=h_1^\ast=0.3366,~h_2=h_2^\ast=0.2310$, red line is with $h_1=0.45,~h_2=0.1$, and blue line is with $h_1=0.1,~h_2=0.25$.
Parameter values for Fig.1 and Fig.2
 Parameter Value $r_1$ 0.5 $r_2$ 0.4 $h_1$ 0.02 $h_2$ 0.055 $a_{11}$ 0.5 $a_{12}$ 0.2 $a_{21}$ 0.25 $a_{22}$ 0.4 $\tau_1$ 10 $\tau_2$ 8 $\sigma_1$ 0.4 $\sigma_2$ 0.3
 Parameter Value $r_1$ 0.5 $r_2$ 0.4 $h_1$ 0.02 $h_2$ 0.055 $a_{11}$ 0.5 $a_{12}$ 0.2 $a_{21}$ 0.25 $a_{22}$ 0.4 $\tau_1$ 10 $\tau_2$ 8 $\sigma_1$ 0.4 $\sigma_2$ 0.3
Parameter values for Fig.3
 Parameter Value $r_1$ 0.8 $r_2$ 0.5 $a_{11}$ 0.5 $a_{12}$ 0.2 $a_{21}$ 0.1 $a_{22}$ 0.4 $\tau_1$ 10 $\tau_2$ 10 $\sigma_1^2$ 0.2 $\sigma_2^2$ 0.2
 Parameter Value $r_1$ 0.8 $r_2$ 0.5 $a_{11}$ 0.5 $a_{12}$ 0.2 $a_{21}$ 0.1 $a_{22}$ 0.4 $\tau_1$ 10 $\tau_2$ 10 $\sigma_1^2$ 0.2 $\sigma_2^2$ 0.2
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