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September  2016, 21(7): 2129-2143. doi: 10.3934/dcdsb.2016040

Semi-Kolmogorov models for predation with indirect effects in random environments

Tomás Caraballo 1, , Renato Colucci 2, and  Xiaoying Han 3,

1. 

Dpto. Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Campus Reina Mercedes, Apdo. de Correos 1160, 41080 Sevilla
2. 

Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080 Sevilla, Spain
3. 

221 Parker Hall, Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849

Received  September 2015 Revised  February 2016 Published  August 2016

In this work we study semi-Kolmogorov models for predation with both the carrying capacities and the indirect effects varying with respect to randomly fluctuating environments. In particular, we consider one random semi-Kolmogorov system involving random and essentially bounded parameters, and one stochastic semi-Kolmogorov system involving white noise and stochastic parameters defined upon a continuous-time Markov chain. For both systems we investigate the existence and uniqueness of solutions, as well as positiveness and boundedness of solutions. For the random semi-Kolmogorov system we also obtain sufficient conditions for the existence of a global random attractor.
Keywords: Kolmogorov predator-prey model, population dynamics, random attractor, Random dynamical system, regime-switching diffusion., indirect effect.
Mathematics Subject Classification: Primary: 92D25; Secondary: 34C6.
Citation: Tomás Caraballo, Renato Colucci, Xiaoying Han. Semi-Kolmogorov models for predation with indirect effects in random environments. Discrete & Continuous Dynamical Systems - B, 2016, 21 (7) : 2129-2143. doi: 10.3934/dcdsb.2016040
References:
[1]

L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar

[2]

Y. Asai and P. E. Kloeden, Numerical schemes for random ODEs via stochastic differential equations, Commun. Appl. Analysis, 17 (2013), 521-528.  Google Scholar

[3]

J. Bao, X. Mao, G. Yin and C. Yuan, Competitive Lotka-Volterra population dynamics with jumps, Nonlinear Analysis: Theory, Methods & Applications, 74 (2011), 6601-6616. doi: 10.1016/j.na.2011.06.043.  Google Scholar

[4]

B. Bolker, M. Holyoak, V. Krivan, L. Rowe and O. Schmitz, Connecting theoretical and empirical studies of trait-mediated interactions, Ecology, 84 (2003), 1101-1114. Google Scholar

[5]

J. L. Brooks and I. D. Stanley, Predation, body size, and composition of plankton, Science, 150 (1965), 28-35. doi: 10.1126/science.150.3692.28.  Google Scholar

[6]

D. Cariveau, R. E. Irwin, A. K. Brody, S. L. Garcia-Mayeya and A. Von der Ohe, Direct and indirect effects of pollinators and seed predators to selection on plant and floral traits, OIKOS, 104 (2004), 15-26. doi: 10.1111/j.0030-1299.2004.12641.x.  Google Scholar

[7]

T. Caraballo, R. Colucci and X. Han, Non-autonomous dynamics of a semi-kolmogorov population model with periodic forcing, Nonlinear Anal. Real World Appl., 31 (2016), 661-680. doi: 10.1016/j.nonrwa.2016.03.007.  Google Scholar

[8]

T. Caraballo, R. Colucci and X. Han, Predation with indirect effects in Fluctuating Environments, Nonlinear Dynamics, 84 (2016), 115-126. doi: 10.1007/s11071-015-2238-3.  Google Scholar

[9]

T. Caraballo and K. Lu, Attractors for stochastic lattice dynamical systems with a multiplicative noise, Front. Math. China, 3 (2008), 317-335. doi: 10.1007/s11464-008-0028-7.  Google Scholar

[10]

T. Caraballo, G. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact nonautonomous dynamical systems, Nonlinear Analysis TMA, 64 (2006), 484-498. Google Scholar

[11]

J. E. Cohen, T. Luczak, C. M. Newman and Z. M. Zhou, Stochastic structure and nonlinear dynamics of food webs: qualitative stability in a lotka-volterra cascade model, Proceedings of the Royal Society of London. Series B, Biological Sciences, 240 (1990), 607-627. doi: 10.1098/rspb.1990.0055.  Google Scholar

[12]

R. Colucci, Coexistence in a one-predator, two-prey system with indirect effects, Journal of Applied Mathematics, (2013), Article ID 625391, 13 pages.  Google Scholar

[13]

R. Colucci and D. Nunez, Periodic orbits for a three-dimensional biological differential systems, Abstract and Applied Analysis, (2013), Article ID 465183, 10 pages.  Google Scholar

[14]

H. Crauel and P. E. Kloeden, Nonautonomous and random attractors, Jahresber Dtsch Math-Ver, 117 (2015), 173-206. doi: 10.1365/s13291-015-0115-0.  Google Scholar

[15]

N. H. Dang, N. H. Du and G. Yin, Existence of stationary distributions for Kolmogorov systems of competivey type under telegraph noise, J. Diff. Equ., 257 (2014), 2078-2101. doi: 10.1016/j.jde.2014.05.029.  Google Scholar

[16]

N. H. Du, R. Kon, K. Sato and Y. Takeuchi, Dynamical behavior of Lotka-Volterra competition systems: Nonautonomous bistable case and the effect of telegraph noise, J. Comput. Appl. Math., 170 (2004), 399-422. doi: 10.1016/j.cam.2004.02.001.  Google Scholar

[17]

, Indirect effects affect ecosystem dynamics., , (2011).   Google Scholar

[18]

F. Flandoli and B. Schmalfuß, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise, Stochastics Stochastics Rep., 59 (1996), 21-45. doi: 10.1080/17442509608834083.  Google Scholar

[19]

J. Hulsman and F. J. Weissing, Biodiversity of Plankton by species oscillations and Chaos, Nature, 402 (1999). Google Scholar

[20]

C. Jeffries, Stability of predation ecosystem models, Ecology, 57 (1976), 1321-1325. doi: 10.2307/1935058.  Google Scholar

[21]

D. Jiang, Ningzhong 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.  Google Scholar

[22]

Q. Luo and X. Mao, Stochastic population dynamics under regime switching, J. Math. Anal. Appl., 334 (2007), 69-84. doi: 10.1016/j.jmaa.2006.12.032.  Google Scholar

[23]

Q Luo and X. Mao, Stochastic population dynamics under regime switching II, J. Math. Anal. Appl., 355 (2009), 577-593. doi: 10.1016/j.jmaa.2009.02.010.  Google Scholar

[24]

P. E. Kloeden and E. Platen, Numerical Solutions to Stochastic Differential Equations, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-12616-5.  Google Scholar

[25]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, American Mathematical Society, Providence, RI, 2011. doi: 10.1090/surv/176.  Google Scholar

[26]

X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, 2006. doi: 10.1142/p473.  Google Scholar

[27]

B. A. Menge, Indirect effects in marine rocky intertidal interaction webs: Patterns and importance, Ecological Monographs, 65 (1995), 21-74. doi: 10.2307/2937158.  Google Scholar

[28]

K. Rohde, Nonequilibrium Ecology, Cambridge University Press, 2005. doi: 10.1017/CBO9780511542152.  Google Scholar

[29]

M. Slatkin, The dynamics of a population in a Markovian environment, Ecology, 59 (1978), 249-256. doi: 10.2307/1936370.  Google Scholar

[30]

Y. Takeuchi, N. H. Du, N. T. Hieu and K. Sato, Evolution of predator-prey systems described by a Lotka-Volterra equation under random environment, J. Math. Anal. Appl., 323 (2006), 938-957. doi: 10.1016/j.jmaa.2005.11.009.  Google Scholar

[31]

M. R. Walsh and D. N. Reznick, Interactions between the direct and indirect effects of predators determine life history evolution in a killifish, Pnas,, www.pnas.org/cgi/doi/10.1073/pnas.0710051105., ().   Google Scholar

[32]

J. T. Wootton, Indirect effects, prey susceptibility, and habitat selection: Impacts of birds on limpets and algae, Ecology, 73 (1992), 981-991. doi: 10.2307/1940174.  Google Scholar

[33]

F. Wu, S. Hu and Y. Liu, Positive solution and its asymptotic behaviour of stochastic functional Kolmogorov-type system, J. Math. Anal. Appl., 364 (2010), 104-118. doi: 10.1016/j.jmaa.2009.10.072.  Google Scholar

[34]

F. Wu and Y. Xu, Stochastic Lotka-Volterra population dynamics with infinite delay, SIAM J. Appl. Math., 70 (2009), 641-657. doi: 10.1137/080719194.  Google Scholar

[35]

C. Zhu and G. Yin, On hybrid competitive Lotka-Volterra ecosystems, Nonlinear Analysis: Theory, Methods & Applications, 71 (2009), e1370-e1379. doi: 10.1016/j.na.2009.01.166.  Google Scholar

show all references

References:
[1]

L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar

[2]

Y. Asai and P. E. Kloeden, Numerical schemes for random ODEs via stochastic differential equations, Commun. Appl. Analysis, 17 (2013), 521-528.  Google Scholar

[3]

J. Bao, X. Mao, G. Yin and C. Yuan, Competitive Lotka-Volterra population dynamics with jumps, Nonlinear Analysis: Theory, Methods & Applications, 74 (2011), 6601-6616. doi: 10.1016/j.na.2011.06.043.  Google Scholar

[4]

B. Bolker, M. Holyoak, V. Krivan, L. Rowe and O. Schmitz, Connecting theoretical and empirical studies of trait-mediated interactions, Ecology, 84 (2003), 1101-1114. Google Scholar

[5]

J. L. Brooks and I. D. Stanley, Predation, body size, and composition of plankton, Science, 150 (1965), 28-35. doi: 10.1126/science.150.3692.28.  Google Scholar

[6]

D. Cariveau, R. E. Irwin, A. K. Brody, S. L. Garcia-Mayeya and A. Von der Ohe, Direct and indirect effects of pollinators and seed predators to selection on plant and floral traits, OIKOS, 104 (2004), 15-26. doi: 10.1111/j.0030-1299.2004.12641.x.  Google Scholar

[7]

T. Caraballo, R. Colucci and X. Han, Non-autonomous dynamics of a semi-kolmogorov population model with periodic forcing, Nonlinear Anal. Real World Appl., 31 (2016), 661-680. doi: 10.1016/j.nonrwa.2016.03.007.  Google Scholar

[8]

T. Caraballo, R. Colucci and X. Han, Predation with indirect effects in Fluctuating Environments, Nonlinear Dynamics, 84 (2016), 115-126. doi: 10.1007/s11071-015-2238-3.  Google Scholar

[9]

T. Caraballo and K. Lu, Attractors for stochastic lattice dynamical systems with a multiplicative noise, Front. Math. China, 3 (2008), 317-335. doi: 10.1007/s11464-008-0028-7.  Google Scholar

[10]

T. Caraballo, G. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact nonautonomous dynamical systems, Nonlinear Analysis TMA, 64 (2006), 484-498. Google Scholar

[11]

J. E. Cohen, T. Luczak, C. M. Newman and Z. M. Zhou, Stochastic structure and nonlinear dynamics of food webs: qualitative stability in a lotka-volterra cascade model, Proceedings of the Royal Society of London. Series B, Biological Sciences, 240 (1990), 607-627. doi: 10.1098/rspb.1990.0055.  Google Scholar

[12]

R. Colucci, Coexistence in a one-predator, two-prey system with indirect effects, Journal of Applied Mathematics, (2013), Article ID 625391, 13 pages.  Google Scholar

[13]

R. Colucci and D. Nunez, Periodic orbits for a three-dimensional biological differential systems, Abstract and Applied Analysis, (2013), Article ID 465183, 10 pages.  Google Scholar

[14]

H. Crauel and P. E. Kloeden, Nonautonomous and random attractors, Jahresber Dtsch Math-Ver, 117 (2015), 173-206. doi: 10.1365/s13291-015-0115-0.  Google Scholar

[15]

N. H. Dang, N. H. Du and G. Yin, Existence of stationary distributions for Kolmogorov systems of competivey type under telegraph noise, J. Diff. Equ., 257 (2014), 2078-2101. doi: 10.1016/j.jde.2014.05.029.  Google Scholar

[16]

N. H. Du, R. Kon, K. Sato and Y. Takeuchi, Dynamical behavior of Lotka-Volterra competition systems: Nonautonomous bistable case and the effect of telegraph noise, J. Comput. Appl. Math., 170 (2004), 399-422. doi: 10.1016/j.cam.2004.02.001.  Google Scholar

[17]

, Indirect effects affect ecosystem dynamics., , (2011).   Google Scholar

[18]

F. Flandoli and B. Schmalfuß, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise, Stochastics Stochastics Rep., 59 (1996), 21-45. doi: 10.1080/17442509608834083.  Google Scholar

[19]

J. Hulsman and F. J. Weissing, Biodiversity of Plankton by species oscillations and Chaos, Nature, 402 (1999). Google Scholar

[20]

C. Jeffries, Stability of predation ecosystem models, Ecology, 57 (1976), 1321-1325. doi: 10.2307/1935058.  Google Scholar

[21]

D. Jiang, Ningzhong 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.  Google Scholar

[22]

Q. Luo and X. Mao, Stochastic population dynamics under regime switching, J. Math. Anal. Appl., 334 (2007), 69-84. doi: 10.1016/j.jmaa.2006.12.032.  Google Scholar

[23]

Q Luo and X. Mao, Stochastic population dynamics under regime switching II, J. Math. Anal. Appl., 355 (2009), 577-593. doi: 10.1016/j.jmaa.2009.02.010.  Google Scholar

[24]

P. E. Kloeden and E. Platen, Numerical Solutions to Stochastic Differential Equations, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-12616-5.  Google Scholar

[25]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, American Mathematical Society, Providence, RI, 2011. doi: 10.1090/surv/176.  Google Scholar

[26]

X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, 2006. doi: 10.1142/p473.  Google Scholar

[27]

B. A. Menge, Indirect effects in marine rocky intertidal interaction webs: Patterns and importance, Ecological Monographs, 65 (1995), 21-74. doi: 10.2307/2937158.  Google Scholar

[28]

K. Rohde, Nonequilibrium Ecology, Cambridge University Press, 2005. doi: 10.1017/CBO9780511542152.  Google Scholar

[29]

M. Slatkin, The dynamics of a population in a Markovian environment, Ecology, 59 (1978), 249-256. doi: 10.2307/1936370.  Google Scholar

[30]

Y. Takeuchi, N. H. Du, N. T. Hieu and K. Sato, Evolution of predator-prey systems described by a Lotka-Volterra equation under random environment, J. Math. Anal. Appl., 323 (2006), 938-957. doi: 10.1016/j.jmaa.2005.11.009.  Google Scholar

[31]

M. R. Walsh and D. N. Reznick, Interactions between the direct and indirect effects of predators determine life history evolution in a killifish, Pnas,, www.pnas.org/cgi/doi/10.1073/pnas.0710051105., ().   Google Scholar

[32]

J. T. Wootton, Indirect effects, prey susceptibility, and habitat selection: Impacts of birds on limpets and algae, Ecology, 73 (1992), 981-991. doi: 10.2307/1940174.  Google Scholar

[33]

F. Wu, S. Hu and Y. Liu, Positive solution and its asymptotic behaviour of stochastic functional Kolmogorov-type system, J. Math. Anal. Appl., 364 (2010), 104-118. doi: 10.1016/j.jmaa.2009.10.072.  Google Scholar

[34]

F. Wu and Y. Xu, Stochastic Lotka-Volterra population dynamics with infinite delay, SIAM J. Appl. Math., 70 (2009), 641-657. doi: 10.1137/080719194.  Google Scholar

[35]

C. Zhu and G. Yin, On hybrid competitive Lotka-Volterra ecosystems, Nonlinear Analysis: Theory, Methods & Applications, 71 (2009), e1370-e1379. doi: 10.1016/j.na.2009.01.166.  Google Scholar

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