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