January  2021, 26(1): 633-644. doi: 10.3934/dcdsb.2020374

The $ P^* $ rule in the stochastic Holt-Lawton model of apparent competition

Department of Evolution and Ecology, and Center for Population Biology, University of California, Davis, CA, 95616, USA

Received  July 2020 Revised  November 2020 Published  December 2020

Fund Project: The author is supported by U.S. National Science Foundation grant DMS-1716803

In 1993, Holt and Lawton introduced a stochastic model of two host species parasitized by a common parasitoid species. We introduce and analyze a generalization of these stochastic difference equations with any number of host species, stochastically varying parasitism rates, stochastically varying host intrinsic fitnesses, and stochastic immigration of parasitoids. Despite the lack of direct, host density-dependence, we show that this system is dissipative i.e. enters a compact set in finite time for all initial conditions. When there is a single host species, stochastic persistence and extinction of the host is characterized using external Lyapunov exponents corresponding to the average per-capita growth rates of the host when rare. When a single host persists, say species $ i $, a explicit expression is derived for the average density, $ P_i^* $, of the parasitoid at the stationary distributions supporting both species. When there are multiple host species, we prove that the host species with the largest $ P_i^* $ value stochastically persists, while the other host species are asymptotically driven to extinction. A review of the main mathematical methods used to prove the results and future challenges are given.

Citation: Sebastian J. Schreiber. The $ P^* $ rule in the stochastic Holt-Lawton model of apparent competition. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 633-644. doi: 10.3934/dcdsb.2020374
References:
[1]

A. S. AcklehY. M. Dib and S. R. J. Jang, Competitive exclusion and coexistence in a nonlinear refuge-mediated selection model, Discrete and Continuous Dynamical Systems–Series B, 7 (2007), 683-698.  doi: 10.3934/dcdsb.2007.7.683.  Google Scholar

[2]

A. S. Ackleh and L. J. S. Allen, Competitive exclusion in SIS and SIR epidemic models with total cross immunity and density-dependent host mortality, Discrete and Continuous Dynamical Systems–Series B, 5 (2005), 175-188.  doi: 10.3934/dcdsb.2005.5.175.  Google Scholar

[3]

M. Benaïm and S. J. Schreiber, Persistence and extinction for stochastic ecological models with internal and external variables, Journal of Mathematical Biology, 79 (2019), 393-431.  doi: 10.1007/s00285-019-01361-4.  Google Scholar

[4]

T. Black, Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals, Discrete and Continuous Dynamical Systems–Series B, 22 (2017), 1253-1272.  doi: 10.3934/dcdsb.2017061.  Google Scholar

[5]

M. B. Bonsall and M. P. Hassell, Apparent competition structures ecological assemblages, Nature, 388 (1997), 371-373.  doi: 10.1038/41084.  Google Scholar

[6]

P. L. Chesson, The stabilizing effect of a random environment, Journal of Mathematical Biology, 15 (1982), 1-36.  doi: 10.1007/BF00275786.  Google Scholar

[7]

F. GrognardF. Mazenc and A. Rapaport, Polytopic Lyapunov functions for persistence analysis of competing species, Discrete and Continuous Dynamical Systems–Series B, 8 (2007), 73-93.  doi: 10.3934/dcdsb.2007.8.73.  Google Scholar

[8]

M. P. Hassell, Host–parasitoid population dynamics, Journal of Animal Ecology, 69 (2000), 543-566.  doi: 10.1046/j.1365-2656.2000.00445.x.  Google Scholar

[9]

M. P. Hassell, The Dynamics of Arthropod Predator-Prey Systems, Monographs in Population Biology, vol. 13, Princeton University Press, Princeton, NJ, 1978.  Google Scholar

[10]

M. P. HassellJ. H. Lawton and J. R. Beddington, Sigmoid functional responses by invertebrate predators and parasitoids, The Journal of Animal Ecology, 46 (1977), 249-262.  doi: 10.2307/3959.  Google Scholar

[11]

A. Hening, D. H. Nguyen and P. Chesson, A general theory of coexistence and extinction for stochastic ecological communities, preprint, 2020, arXiv: 2007.09025. Google Scholar

[12]

J. Hofbauer, A general cooperation theorem for hypercycles, Monatshefte für Mathematik, 91 (1981), 233-240.  doi: 10.1007/BF01301790.  Google Scholar

[13] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998.  doi: 10.1017/CBO9781139173179.  Google Scholar
[14]

R. D. Holt, Ecology at the mesoscale: The influence of regional processes on local communities, In Species Diversity in Ecological Communities, University of Chicago Press, Chicago, 1993, 77-88. Google Scholar

[15]

R. D. Holt, Predation, apparent competition and the structure of prey communities, Theoretical Population Biology, 12 (1977), 197-229.  doi: 10.1016/0040-5809(77)90042-9.  Google Scholar

[16]

R. D. Holt and M. B. Bonsall, Apparent competition, Annual Review of Ecology, Evolution, and Systematics, 48 (2017), 447-471.  doi: 10.1146/annurev-ecolsys-110316-022628.  Google Scholar

[17]

R. D. Holt and J. H. Lawton, Apparent competition and enemy-free space in insect host-parasitoid communities, American Naturalist, 142 (1993), 623-645.  doi: 10.1086/285561.  Google Scholar

[18]

S. B. HsuM. C. LiW. Liu and M. Malkin, Heteroclinic foliation, global oscillations for the Nicholson-Bailey model and delay of stability loss, Discrete and Continous Dynamical Systems, 9 (2003), 1465-1492.  doi: 10.3934/dcds.2003.9.1465.  Google Scholar

[19]

S. B. Hsu and C. J. Lin, Dynamics of two phytoplankton species competing for light and nutrient with internal storage, Discrete and Continuous Dynamical Systems–Series S, 7 (2014), 1259-1285.  doi: 10.3934/dcdss.2014.7.1259.  Google Scholar

[20]

S. B. Hsu and L. W. Roeger, A refuge-mediated apparent competition model, Canadian Applied Mathematical Quarterly, 19 (2011), 219-234.   Google Scholar

[21]

S. B. HsuJ. Shi and and F. B. Wang, Further studies of a reaction-diffusion system for an unstirred chemostat with internal storage, Discrete and Continuous Dynamical Systems–Series B, 19 (2014), 3169-3189.  doi: 10.3934/dcdsb.2014.19.3169.  Google Scholar

[22]

T. B. Issa and R. B. Salako, Asymptotic dynamics in a two-species chemotaxis model with non-local terms, Discrete and Continuous Dynamical Systems–Series B, 22 (2017), 3839-3874.  doi: 10.3934/dcdsb.2017193.  Google Scholar

[23]

W. T. Jamieson and J. Reis, Global behavior for the classical Nicholson–Bailey model, Journal of Mathematical Analysis and Applications, 461 (2018), 492-499.  doi: 10.1016/j.jmaa.2017.12.071.  Google Scholar

[24] A. Katok and B. Hasselblatt, Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995.   Google Scholar
[25]

Y. KuangJ. D. Nagy and J. J. Elser, Biological stoichiometry of tumor dynamics: Mathematical models and analysis, Discrete and Continuous Dynamical Systems–Series B, 4 (2004), 221-240.  doi: 10.3934/dcdsb.2004.4.221.  Google Scholar

[26]

M. R. S. Kulenović and O. Merino, Competitive-exclusion versus competitive-coexistence for systems in the plane, Discrete and Continuous Dynamical Systems–Series B., 6 (2006), 1141-1156.  doi: 10.3934/dcdsb.2006.6.1141.  Google Scholar

[27]

C. J. Lin, Competition of two phytoplankton species for light with wavelength, Discrete and Continuous Dynamical Systems–Series B., 21 (2016), 523-536.  doi: 10.3934/dcdsb.2016.21.523.  Google Scholar

[28]

J. Loman, A graphical solution to a one-predator, two-prey system with apparent competition and mutualism, Mathematical Biosciences, 91 (1988), 1-16.  doi: 10.1016/0025-5564(88)90021-1.  Google Scholar

[29]

Y. Lou and D. Munther, Dynamics of a three species competition model, Discrete and Continuous Dynamical Systems–Series B, 32 (2012), 3099-3131.  doi: 10.3934/dcds.2012.32.3099.  Google Scholar

[30]

W. T. Jamieson and J. Reis, Global behavior for the classical Nicholson–Bailey model, Journal of Mathematical Analysis and Applications, 461 (2018), 492-499.  doi: 10.1016/j.jmaa.2017.12.071.  Google Scholar

[31]

R. M. May, Host-parasitoid systems in patch environments: A phenomenological model, Journal of Animal Ecology, 47 (1978), 833-843.   Google Scholar

[32]

R. M. MayM. P. HassellR. M. Anderson and D. W. Tonkyn, Density dependence in host-parasitoid models, Journal of Animal Ecology, 50 (1981), 855-865.  doi: 10.2307/4142.  Google Scholar

[33]

C. R. MillerY. KuangW. F. Fagan and J. J. Elser, Modeling and analysis of stoichiometric two-patch consumer-resource systems, Mathematical Biosciences, 189 (2004), 153-184.  doi: 10.1016/j.mbs.2004.01.004.  Google Scholar

[34]

N. J. Mills and W. M. Getz, Modelling the biological control of insect pests: A review of host-parasitoid models, Ecological Modelling, 92 (1996), 121-143.  doi: 10.1016/0304-3800(95)00177-8.  Google Scholar

[35]

R. J. MorrisO. T. Lewis and H. C. J. Godfray, Experimental evidence for apparent competition in a tropical forest food web, Nature, 428 (2004), 310-313.  doi: 10.1038/nature02394.  Google Scholar

[36]

T. Namba, Dispersal-mediated coexistence of indirect competitors in source-sink metacommunities, Japan Journal of Industrial and Applied Mathematics, 24 (2007), 39-55.  doi: 10.1007/BF03167506.  Google Scholar

[37]

A. J. Nicholson and V. A. Bailey, The balance of animal populations, Proceedings of the Zoological Society of London, (1935), 551–598. Google Scholar

[38]

M. Núñez LópezJ. X. Velasco-Hernández and P. A. Marquet, The dynamics of technological change under constraints: Adopters and resources, Discrete and Continuous Dynamical Systems–Series B, 19 (2014), 3299-3317.  doi: 10.3934/dcdsb.2014.19.3299.  Google Scholar

[39]

H. PourbashashS. S. PilyuginP. De Leenheer and C. McCluskey, Global analysis of within host virus models with cell-to-cell viral transmission, Discrete and Continuous Dynamical Systems–Series B, 19 (2014), 3341-3357.  doi: 10.3934/dcdsb.2014.19.3341.  Google Scholar

[40]

A. Rapaport and M. Veruete, A new proof of the competitive exclusion principle in the chemostat, Discrete and Continuous Dynamical Systems–Series B, 24 (2019), 3755-3764.  doi: 10.3934/dcdsb.2018314.  Google Scholar

[41]

T. W. Schoener, The newest synthesis: Understanding the interplay of evolutionary and ecological dynamics, Science, 331 (2011), 426-429.  doi: 10.1126/science.1193954.  Google Scholar

[42]

S. J. Schreiber, Coexistence for species sharing a predator, Journal of Differential Equations, 196 (2004), 209-225.  doi: 10.1016/S0022-0396(03)00169-4.  Google Scholar

[43]

S. J. Schreiber, Persistence for stochastic difference equations: A mini-review, Journal of Difference Equations and Applications, 18 (2012), 1381-1403.  doi: 10.1080/10236198.2011.628662.  Google Scholar

[44]

S. J. Schreiber, When do factors promoting genetic diversity also promote population persistence? A demographic perspective on Gillespieś SAS-CFF model, Theoretical Population Biology, 133 (2020), 141-149.   Google Scholar

[45]

S. J. Schreiber, Coexistence in the Face of Uncertainty, Recent Progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science, Springer, New York, 2017,349–384.  Google Scholar

[46]

S. J. Schreiber and V. Křivan, Holt (1977) and apparent competition, Theoretical Population Biology, 133 (2020), 17-18.  doi: 10.1016/j.tpb.2019.09.006.  Google Scholar

[47]

S. J. SchreiberM. Benaïm and K. A. S. Atchadé, Persistence in fluctuating environments, Journal of Mathematical Biology, 62 (2011), 655-683.  doi: 10.1007/s00285-010-0349-5.  Google Scholar

[48]

S. J. SchreiberR. Bürger and D. I. Bolnick, The community effects of phenotypic and genetic variation within a predator population, Ecology, 92 (2011), 1582-1593.  doi: 10.1890/10-2071.1.  Google Scholar

[49]

S. J. Schreiber and S. Patel, Evolutionarily induced alternative states and coexistence in systems with apparent competition, Natural Resource Modelling, 28 (2015), 475-496.  doi: 10.1111/nrm.12076.  Google Scholar

[50]

H. L. Smith and X. Q. Zhao, Competitive exclusion in a discrete-time, size-structured chemostat model, Discrete and Continuous Dynamical Systems–Series B, 1 (2001), 183-191.  doi: 10.3934/dcdsb.2001.1.183.  Google Scholar

[51]

D. Tang, Dynamical behavior for a Lotka-Volterra weak competition system in advective homogeneous environment, Discrete and Continuous Dynamical Systems–Series B, 24 (2019), 4913-4928.  doi: 10.3934/dcdsb.2019037.  Google Scholar

[52]

D. Tilman, Resource Competition and Community Structure, Monographs in Population Biology, vol. 17, Princeton University Press, Princeton, NJ, 1982. Google Scholar

[53]

J. X. Velasco-HernándezM. Núñez LópezG. Ramírez-Santiago and M. Hernández-Rosales, On carrying-capacity construction, metapopulations and density-dependent mortality, Discrete and Continuous Dynamical Systems–Series B, 22 (2017), 1099-1110.  doi: 10.3934/dcdsb.2017054.  Google Scholar

[54]

V. Volterra, Fluctuations in the abundance of a species considered mathematically, Nature, 119 (1927), 12-13.  doi: 10.1038/119012b0.  Google Scholar

[55]

H. WangK. DunningJ. J. Elser and Y. Kuang, Daphnia species invasion, competitive exclusion, and chaotic coexistence, Discrete and Continuous Dynamical Systems–Series B, 12 (2009), 481-493.  doi: 10.3934/dcdsb.2009.12.481.  Google Scholar

[56]

Y. WuN. Tuncer and M. Martcheva, Coexistence and competitive exclusion in an SIS model with standard incidence and diffusion, Discrete and Continuous Dynamical Systems–Series B, 22 (2017), 1167-1187.  doi: 10.3934/dcdsb.2017057.  Google Scholar

[57]

H. YuS. Zhong and R. P. Agarwal, Mathematics and dynamic analysis of an apparent competition community model with impulsive effect, Mathematical and Computer Modelling, 52 (2010), 25-36.  doi: 10.1016/j.mcm.2009.11.019.  Google Scholar

show all references

References:
[1]

A. S. AcklehY. M. Dib and S. R. J. Jang, Competitive exclusion and coexistence in a nonlinear refuge-mediated selection model, Discrete and Continuous Dynamical Systems–Series B, 7 (2007), 683-698.  doi: 10.3934/dcdsb.2007.7.683.  Google Scholar

[2]

A. S. Ackleh and L. J. S. Allen, Competitive exclusion in SIS and SIR epidemic models with total cross immunity and density-dependent host mortality, Discrete and Continuous Dynamical Systems–Series B, 5 (2005), 175-188.  doi: 10.3934/dcdsb.2005.5.175.  Google Scholar

[3]

M. Benaïm and S. J. Schreiber, Persistence and extinction for stochastic ecological models with internal and external variables, Journal of Mathematical Biology, 79 (2019), 393-431.  doi: 10.1007/s00285-019-01361-4.  Google Scholar

[4]

T. Black, Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals, Discrete and Continuous Dynamical Systems–Series B, 22 (2017), 1253-1272.  doi: 10.3934/dcdsb.2017061.  Google Scholar

[5]

M. B. Bonsall and M. P. Hassell, Apparent competition structures ecological assemblages, Nature, 388 (1997), 371-373.  doi: 10.1038/41084.  Google Scholar

[6]

P. L. Chesson, The stabilizing effect of a random environment, Journal of Mathematical Biology, 15 (1982), 1-36.  doi: 10.1007/BF00275786.  Google Scholar

[7]

F. GrognardF. Mazenc and A. Rapaport, Polytopic Lyapunov functions for persistence analysis of competing species, Discrete and Continuous Dynamical Systems–Series B, 8 (2007), 73-93.  doi: 10.3934/dcdsb.2007.8.73.  Google Scholar

[8]

M. P. Hassell, Host–parasitoid population dynamics, Journal of Animal Ecology, 69 (2000), 543-566.  doi: 10.1046/j.1365-2656.2000.00445.x.  Google Scholar

[9]

M. P. Hassell, The Dynamics of Arthropod Predator-Prey Systems, Monographs in Population Biology, vol. 13, Princeton University Press, Princeton, NJ, 1978.  Google Scholar

[10]

M. P. HassellJ. H. Lawton and J. R. Beddington, Sigmoid functional responses by invertebrate predators and parasitoids, The Journal of Animal Ecology, 46 (1977), 249-262.  doi: 10.2307/3959.  Google Scholar

[11]

A. Hening, D. H. Nguyen and P. Chesson, A general theory of coexistence and extinction for stochastic ecological communities, preprint, 2020, arXiv: 2007.09025. Google Scholar

[12]

J. Hofbauer, A general cooperation theorem for hypercycles, Monatshefte für Mathematik, 91 (1981), 233-240.  doi: 10.1007/BF01301790.  Google Scholar

[13] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998.  doi: 10.1017/CBO9781139173179.  Google Scholar
[14]

R. D. Holt, Ecology at the mesoscale: The influence of regional processes on local communities, In Species Diversity in Ecological Communities, University of Chicago Press, Chicago, 1993, 77-88. Google Scholar

[15]

R. D. Holt, Predation, apparent competition and the structure of prey communities, Theoretical Population Biology, 12 (1977), 197-229.  doi: 10.1016/0040-5809(77)90042-9.  Google Scholar

[16]

R. D. Holt and M. B. Bonsall, Apparent competition, Annual Review of Ecology, Evolution, and Systematics, 48 (2017), 447-471.  doi: 10.1146/annurev-ecolsys-110316-022628.  Google Scholar

[17]

R. D. Holt and J. H. Lawton, Apparent competition and enemy-free space in insect host-parasitoid communities, American Naturalist, 142 (1993), 623-645.  doi: 10.1086/285561.  Google Scholar

[18]

S. B. HsuM. C. LiW. Liu and M. Malkin, Heteroclinic foliation, global oscillations for the Nicholson-Bailey model and delay of stability loss, Discrete and Continous Dynamical Systems, 9 (2003), 1465-1492.  doi: 10.3934/dcds.2003.9.1465.  Google Scholar

[19]

S. B. Hsu and C. J. Lin, Dynamics of two phytoplankton species competing for light and nutrient with internal storage, Discrete and Continuous Dynamical Systems–Series S, 7 (2014), 1259-1285.  doi: 10.3934/dcdss.2014.7.1259.  Google Scholar

[20]

S. B. Hsu and L. W. Roeger, A refuge-mediated apparent competition model, Canadian Applied Mathematical Quarterly, 19 (2011), 219-234.   Google Scholar

[21]

S. B. HsuJ. Shi and and F. B. Wang, Further studies of a reaction-diffusion system for an unstirred chemostat with internal storage, Discrete and Continuous Dynamical Systems–Series B, 19 (2014), 3169-3189.  doi: 10.3934/dcdsb.2014.19.3169.  Google Scholar

[22]

T. B. Issa and R. B. Salako, Asymptotic dynamics in a two-species chemotaxis model with non-local terms, Discrete and Continuous Dynamical Systems–Series B, 22 (2017), 3839-3874.  doi: 10.3934/dcdsb.2017193.  Google Scholar

[23]

W. T. Jamieson and J. Reis, Global behavior for the classical Nicholson–Bailey model, Journal of Mathematical Analysis and Applications, 461 (2018), 492-499.  doi: 10.1016/j.jmaa.2017.12.071.  Google Scholar

[24] A. Katok and B. Hasselblatt, Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995.   Google Scholar
[25]

Y. KuangJ. D. Nagy and J. J. Elser, Biological stoichiometry of tumor dynamics: Mathematical models and analysis, Discrete and Continuous Dynamical Systems–Series B, 4 (2004), 221-240.  doi: 10.3934/dcdsb.2004.4.221.  Google Scholar

[26]

M. R. S. Kulenović and O. Merino, Competitive-exclusion versus competitive-coexistence for systems in the plane, Discrete and Continuous Dynamical Systems–Series B., 6 (2006), 1141-1156.  doi: 10.3934/dcdsb.2006.6.1141.  Google Scholar

[27]

C. J. Lin, Competition of two phytoplankton species for light with wavelength, Discrete and Continuous Dynamical Systems–Series B., 21 (2016), 523-536.  doi: 10.3934/dcdsb.2016.21.523.  Google Scholar

[28]

J. Loman, A graphical solution to a one-predator, two-prey system with apparent competition and mutualism, Mathematical Biosciences, 91 (1988), 1-16.  doi: 10.1016/0025-5564(88)90021-1.  Google Scholar

[29]

Y. Lou and D. Munther, Dynamics of a three species competition model, Discrete and Continuous Dynamical Systems–Series B, 32 (2012), 3099-3131.  doi: 10.3934/dcds.2012.32.3099.  Google Scholar

[30]

W. T. Jamieson and J. Reis, Global behavior for the classical Nicholson–Bailey model, Journal of Mathematical Analysis and Applications, 461 (2018), 492-499.  doi: 10.1016/j.jmaa.2017.12.071.  Google Scholar

[31]

R. M. May, Host-parasitoid systems in patch environments: A phenomenological model, Journal of Animal Ecology, 47 (1978), 833-843.   Google Scholar

[32]

R. M. MayM. P. HassellR. M. Anderson and D. W. Tonkyn, Density dependence in host-parasitoid models, Journal of Animal Ecology, 50 (1981), 855-865.  doi: 10.2307/4142.  Google Scholar

[33]

C. R. MillerY. KuangW. F. Fagan and J. J. Elser, Modeling and analysis of stoichiometric two-patch consumer-resource systems, Mathematical Biosciences, 189 (2004), 153-184.  doi: 10.1016/j.mbs.2004.01.004.  Google Scholar

[34]

N. J. Mills and W. M. Getz, Modelling the biological control of insect pests: A review of host-parasitoid models, Ecological Modelling, 92 (1996), 121-143.  doi: 10.1016/0304-3800(95)00177-8.  Google Scholar

[35]

R. J. MorrisO. T. Lewis and H. C. J. Godfray, Experimental evidence for apparent competition in a tropical forest food web, Nature, 428 (2004), 310-313.  doi: 10.1038/nature02394.  Google Scholar

[36]

T. Namba, Dispersal-mediated coexistence of indirect competitors in source-sink metacommunities, Japan Journal of Industrial and Applied Mathematics, 24 (2007), 39-55.  doi: 10.1007/BF03167506.  Google Scholar

[37]

A. J. Nicholson and V. A. Bailey, The balance of animal populations, Proceedings of the Zoological Society of London, (1935), 551–598. Google Scholar

[38]

M. Núñez LópezJ. X. Velasco-Hernández and P. A. Marquet, The dynamics of technological change under constraints: Adopters and resources, Discrete and Continuous Dynamical Systems–Series B, 19 (2014), 3299-3317.  doi: 10.3934/dcdsb.2014.19.3299.  Google Scholar

[39]

H. PourbashashS. S. PilyuginP. De Leenheer and C. McCluskey, Global analysis of within host virus models with cell-to-cell viral transmission, Discrete and Continuous Dynamical Systems–Series B, 19 (2014), 3341-3357.  doi: 10.3934/dcdsb.2014.19.3341.  Google Scholar

[40]

A. Rapaport and M. Veruete, A new proof of the competitive exclusion principle in the chemostat, Discrete and Continuous Dynamical Systems–Series B, 24 (2019), 3755-3764.  doi: 10.3934/dcdsb.2018314.  Google Scholar

[41]

T. W. Schoener, The newest synthesis: Understanding the interplay of evolutionary and ecological dynamics, Science, 331 (2011), 426-429.  doi: 10.1126/science.1193954.  Google Scholar

[42]

S. J. Schreiber, Coexistence for species sharing a predator, Journal of Differential Equations, 196 (2004), 209-225.  doi: 10.1016/S0022-0396(03)00169-4.  Google Scholar

[43]

S. J. Schreiber, Persistence for stochastic difference equations: A mini-review, Journal of Difference Equations and Applications, 18 (2012), 1381-1403.  doi: 10.1080/10236198.2011.628662.  Google Scholar

[44]

S. J. Schreiber, When do factors promoting genetic diversity also promote population persistence? A demographic perspective on Gillespieś SAS-CFF model, Theoretical Population Biology, 133 (2020), 141-149.   Google Scholar

[45]

S. J. Schreiber, Coexistence in the Face of Uncertainty, Recent Progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science, Springer, New York, 2017,349–384.  Google Scholar

[46]

S. J. Schreiber and V. Křivan, Holt (1977) and apparent competition, Theoretical Population Biology, 133 (2020), 17-18.  doi: 10.1016/j.tpb.2019.09.006.  Google Scholar

[47]

S. J. SchreiberM. Benaïm and K. A. S. Atchadé, Persistence in fluctuating environments, Journal of Mathematical Biology, 62 (2011), 655-683.  doi: 10.1007/s00285-010-0349-5.  Google Scholar

[48]

S. J. SchreiberR. Bürger and D. I. Bolnick, The community effects of phenotypic and genetic variation within a predator population, Ecology, 92 (2011), 1582-1593.  doi: 10.1890/10-2071.1.  Google Scholar

[49]

S. J. Schreiber and S. Patel, Evolutionarily induced alternative states and coexistence in systems with apparent competition, Natural Resource Modelling, 28 (2015), 475-496.  doi: 10.1111/nrm.12076.  Google Scholar

[50]

H. L. Smith and X. Q. Zhao, Competitive exclusion in a discrete-time, size-structured chemostat model, Discrete and Continuous Dynamical Systems–Series B, 1 (2001), 183-191.  doi: 10.3934/dcdsb.2001.1.183.  Google Scholar

[51]

D. Tang, Dynamical behavior for a Lotka-Volterra weak competition system in advective homogeneous environment, Discrete and Continuous Dynamical Systems–Series B, 24 (2019), 4913-4928.  doi: 10.3934/dcdsb.2019037.  Google Scholar

[52]

D. Tilman, Resource Competition and Community Structure, Monographs in Population Biology, vol. 17, Princeton University Press, Princeton, NJ, 1982. Google Scholar

[53]

J. X. Velasco-HernándezM. Núñez LópezG. Ramírez-Santiago and M. Hernández-Rosales, On carrying-capacity construction, metapopulations and density-dependent mortality, Discrete and Continuous Dynamical Systems–Series B, 22 (2017), 1099-1110.  doi: 10.3934/dcdsb.2017054.  Google Scholar

[54]

V. Volterra, Fluctuations in the abundance of a species considered mathematically, Nature, 119 (1927), 12-13.  doi: 10.1038/119012b0.  Google Scholar

[55]

H. WangK. DunningJ. J. Elser and Y. Kuang, Daphnia species invasion, competitive exclusion, and chaotic coexistence, Discrete and Continuous Dynamical Systems–Series B, 12 (2009), 481-493.  doi: 10.3934/dcdsb.2009.12.481.  Google Scholar

[56]

Y. WuN. Tuncer and M. Martcheva, Coexistence and competitive exclusion in an SIS model with standard incidence and diffusion, Discrete and Continuous Dynamical Systems–Series B, 22 (2017), 1167-1187.  doi: 10.3934/dcdsb.2017057.  Google Scholar

[57]

H. YuS. Zhong and R. P. Agarwal, Mathematics and dynamic analysis of an apparent competition community model with impulsive effect, Mathematical and Computer Modelling, 52 (2010), 25-36.  doi: 10.1016/j.mcm.2009.11.019.  Google Scholar

Figure 1.  Stochastic persistence and the $ P^* $ rule for the host-parasitoid model (1). In A, there is $ k = 1 $ host species and the condition for stochastic persistence is met. In B, there are $ k = 4 $ host species which only differ in the variance of their $ R_i(t) $ terms. Parameter values: $ a_i(t) = 0.1 $ for all $ i $ and $ t $, $ R_i(t) = 0.9+1.1\beta^R_i(t) $ where $ \beta_i^R(t) $ are $ \beta $ distributed with both scale parameters $ = k+1-i $, and $ I(t) = 0.1+0.9\beta^I(t) $ where $ \beta^I(t) $ are $ \beta $ distributed with both scale parameters $ = 2 $
[1]

Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037

[2]

María J. Garrido-Atienza, Bohdan Maslowski, Jana  Šnupárková. Semilinear stochastic equations with bilinear fractional noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3075-3094. doi: 10.3934/dcdsb.2016088

[3]

Xianming Liu, Guangyue Han. A Wong-Zakai approximation of stochastic differential equations driven by a general semimartingale. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2499-2508. doi: 10.3934/dcdsb.2020192

[4]

Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175

[5]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213

[6]

Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521

[7]

Xiaohu Wang, Dingshi Li, Jun Shen. Wong-Zakai approximations and attractors for stochastic wave equations driven by additive noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2829-2855. doi: 10.3934/dcdsb.2020207

[8]

Chin-Chin Wu. Existence of traveling wavefront for discrete bistable competition model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 973-984. doi: 10.3934/dcdsb.2011.16.973

[9]

Guo-Bao Zhang, Ruyun Ma, Xue-Shi Li. Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 587-608. doi: 10.3934/dcdsb.2018035

[10]

Dan Wei, Shangjiang Guo. Qualitative analysis of a Lotka-Volterra competition-diffusion-advection system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2599-2623. doi: 10.3934/dcdsb.2020197

[11]

J. Frédéric Bonnans, Justina Gianatti, Francisco J. Silva. On the convergence of the Sakawa-Shindo algorithm in stochastic control. Mathematical Control & Related Fields, 2016, 6 (3) : 391-406. doi: 10.3934/mcrf.2016008

[12]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437

[13]

Seung-Yeal Ha, Dongnam Ko, Chanho Min, Xiongtao Zhang. Emergent collective behaviors of stochastic kuramoto oscillators. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1059-1081. doi: 10.3934/dcdsb.2019208

[14]

Sergi Simon. Linearised higher variational equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4827-4854. doi: 10.3934/dcds.2014.34.4827

[15]

Ardeshir Ahmadi, Hamed Davari-Ardakani. A multistage stochastic programming framework for cardinality constrained portfolio optimization. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 359-377. doi: 10.3934/naco.2017023

[16]

Shangzhi Li, Shangjiang Guo. Permanence and extinction of a stochastic SIS epidemic model with three independent Brownian motions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2693-2719. doi: 10.3934/dcdsb.2020201

[17]

Longxiang Fang, Narayanaswamy Balakrishnan, Wenyu Huang. Stochastic comparisons of parallel systems with scale proportional hazards components equipped with starting devices. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021004

[18]

Shihu Li, Wei Liu, Yingchao Xie. Large deviations for stochastic 3D Leray-$ \alpha $ model with fractional dissipation. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2491-2509. doi: 10.3934/cpaa.2019113

[19]

Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277

[20]

Matthias Erbar, Jan Maas. Gradient flow structures for discrete porous medium equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1355-1374. doi: 10.3934/dcds.2014.34.1355

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (85)
  • HTML views (110)
  • Cited by (0)

Other articles
by authors

[Back to Top]