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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 and 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.

[2]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[12]

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

[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.

[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.

[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.

[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.

[17]

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

[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.

[19]

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

[20]

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

[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.

[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.

[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.

[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.

[25]

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

[26]

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

[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.

[28]

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

[29]

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

[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.

[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., (). 

[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.

[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.

[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.

[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.

show all references

References:
[1]

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

[2]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[12]

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

[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.

[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.

[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.

[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.

[17]

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

[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.

[19]

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

[20]

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

[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.

[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.

[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.

[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.

[25]

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

[26]

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

[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.

[28]

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

[29]

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

[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.

[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., (). 

[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.

[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.

[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.

[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.

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