Keep, break and breakout in food chains with two and three species

Maoxiang Wang; Fenglan Hu; Meng Xu; Zhipeng Qiu; Maoxiang Wang; Fenglan Hu; Meng Xu; Zhipeng Qiu

doi:10.3934/mbe.2021043

Mathematical Biosciences and Engineering

2021, Volume 18, Issue 1: 817-836. doi: 10.3934/mbe.2021043

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Research article

Keep, break and breakout in food chains with two and three species

School of Science, Nanjing University of Science and Technology, Nanjing 210094, China

Received: 20 September 2020 Accepted: 07 December 2020 Published: 23 December 2020

In this paper, through Rosenzweig-MacArthur predator-prey model we study the cyclic coexistence and stationary coexistence and discuss temporal keep and break in the food chain with two species. Then species' diffusion is considered and its effect on oscillation and stability of the ODE system is studied concerning the two different states of coexistence. We find in cyclic coexistence temporal oscillation of population is translated into spatial oscillation although there is fluctuation at the beginning of population waves and finally more stable population evolution is observed. Furthermore, the presence of spatial diffusion of the species can lead to steady wavefront propagation and alter the population distribution in the food chain with two and three species. We show that lower-level species with slow propagation will limit higher-level species and help to keep food chain in space, but through fast propagation lower-level species can survive in a new space without predation and realize a breakout in the linear food chain.
- Rosenzweig-MacArthur model,
- predator-prey,
- food chain,
- wavefront propagation
Citation: Maoxiang Wang, Fenglan Hu, Meng Xu, Zhipeng Qiu. Keep, break and breakout in food chains with two and three species[J]. Mathematical Biosciences and Engineering, 2021, 18(1): 817-836. doi: 10.3934/mbe.2021043

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Abstract

In this paper, through Rosenzweig-MacArthur predator-prey model we study the cyclic coexistence and stationary coexistence and discuss temporal keep and break in the food chain with two species. Then species' diffusion is considered and its effect on oscillation and stability of the ODE system is studied concerning the two different states of coexistence. We find in cyclic coexistence temporal oscillation of population is translated into spatial oscillation although there is fluctuation at the beginning of population waves and finally more stable population evolution is observed. Furthermore, the presence of spatial diffusion of the species can lead to steady wavefront propagation and alter the population distribution in the food chain with two and three species. We show that lower-level species with slow propagation will limit higher-level species and help to keep food chain in space, but through fast propagation lower-level species can survive in a new space without predation and realize a breakout in the linear food chain.

References

[1]	N. J. Gotelli, A. M. Ellison, Food-web models predict species abundances in response to habitat change, PLoS Biol., 4 (2006), 1869-1873.
[2]	S. B. Hsu, T. W. Hwang, Y. Kuang, A Ratio-dependent food chain model and its applications to control, Math. Biosci., 181 (2003), 55-83. doi: 10.1016/S0025-5564(02)00127-X
[3]	C. H. Chiu, S. B. Hsu, Extinction of top-predator in a three-level food-chain model, J. Math. Biol., 37 (1998), 372-380. doi: 10.1007/s002850050134
[4]	K. R. Crooks, M. E. Soulé, Mesopredator release and avifaunal extinctions in a fragmented system, Nature, 400 (1999), 563-566. doi: 10.1038/23028
[5]	J. R. Britton, Introduced parasites in food webs: new species, shifting structures?, Trends Ecol. Evol., 28 (2013), 93-99. doi: 10.1016/j.tree.2012.08.020
[6]	A. J. Lotka, Elements of Physical Biology, Williams and Willkins, Baltimore, 1925.
[7]	V. Volterra, Variations and fluctuations of the number of individuals in animal species living together, ICES. J. Mar. Sci., 3 (1928), 3-51. doi: 10.1093/icesjms/3.1.3
[8]	R. Arditi, L.R. Ginzburg, Coupling in predator-prey dynamics: ratio-dependence, J. Theor. Biol., 139 (1989), 311-326. doi: 10.1016/S0022-5193(89)80211-5
[9]	A. Fenton, M. Spencer, D. J. S. Montagnes, Parameterising variable assimilation efficiency in predator-prey models, Oikos, 119 (2010), 1000-1010. doi: 10.1111/j.1600-0706.2009.17875.x
[10]	J. Li, D. J. Montagnes, Restructuring fundamental predator-prey models by recognizing prey-dependent conversion efficiency and mortality rates, Protist, 166 (2015), 211-223. doi: 10.1016/j.protis.2015.02.003
[11]	T. K. Kar, A. Ghorai, A. Batabyal, Global dynamics and bifurcation of a tri-trophic food chain model, World J. Model. Simul., 8 (2012), 66-80.
[12]	E. E. Joshua, E. T. Akpan, C. E. Madubueze, Hopf-bifurcation limit cycles of an extended Rosenzweig-MacArthur model, J. Math. Res., 8 (2016), 22. doi: 10.5539/jmr.v8n3p22
[13]	B. Deng, Food chain chaos due to junction-fold point, Chaos, 11 (2001), 514. doi: 10.1063/1.1396340
[14]	B. Deng, G. Hines, Food chain chaos due to transcritical point, Chaos, 13 (2003), 578. doi: 10.1063/1.1576531
[15]	D. M. Post, The long and short of food-chain length, Trends Ecol. Evol., 17 (2002), 269. doi: 10.1016/S0169-5347(02)02455-2
[16]	M. Kondon, K. Ninomiya, Food-chain length and adaptive foraging, P. Roy. Soc. B: Biol. Sci., 276 (2009), 3113-3121.
[17]	M. L. Pace, J. J. Cole, S. R. Carpenter, J. F. Kitchell, Trophic cascades revealed in diverse ecosystems, Trends Ecol. Evol., 14 (1999), 483-488. doi: 10.1016/S0169-5347(99)01723-1
[18]	L. Persson, Trophic cascades: Abiding heterogeneity and the trophic level concept at the end of the road, Oikos, 85 (1999), 385-397. doi: 10.2307/3546688
[19]	L. Oksanen, T. Oksanen, The logic and realism of the hypothesis of exploitation ecosystems, Am. Nat., 155 (2000), 703-723. doi: 10.1086/303354
[20]	J. Hofbauer, K. Sigmund, Evolutionary games and population dynamics, Cambridge University Press, Cambridge, 1998.
[21]	C. P. Rocaa, J. A. Cuesta, A. Sanchez, Evolutionary game theory: Temporal and spatial effects beyond replicator dynamics, Phys. Life Rev., 6 (2000), 208.
[22]	D. Melese, S. Gakkhar, Pattern formation in tri-trophic ratio-dependent food chain model, Appl. Math., 2 (2011), 1507-1514. doi: 10.4236/am.2011.212213
[23]	M. C. Cross, P. C. Hohenberg, Pattern formation out of equilibrium, Rev. Mod. Phys., 65 (1993) 851. doi: 10.1103/RevModPhys.65.851
[24]	W. van Saarloos, P. C. Hohenberg, Fronts, pulses, sources, and sinks in generalized complex Ginzburg-Landau equations, Phys. D, 56 (1992), 303. doi: 10.1016/0167-2789(92)90175-M
[25]	S. Amadae, Prisoner's Dilemma, Prisoners of Reason, Cambridge University Press, New York, 2016.
[26]	M. H. Vainstein, J. J. Arenzon, Spatial social dilemmas: Dilution, mobility, and grouping effects with imitation dynamics, Phys. A, 394 (2014), 145. doi: 10.1016/j.physa.2013.09.032
[27]	M. X. Wang, Y. Q. Ma, P. Y. Lai, Regulatory effects on the population dynamics and wave propagation in a cell lineage model, J. Theor. Biol., 393 (2016), 105-117. doi: 10.1016/j.jtbi.2015.12.035
[28]	M. X. Wang, P. Y. Lai, Population dynamics and wave propagation in Lotka-Volterra system with spatial diffusion, Phys. Rev. E, 86 (2012), 051908. doi: 10.1103/PhysRevE.86.051908
[29]	M. X. Wang, Y. Ma, Population evolution in mutualistic Lotka-Volterra system with spatial diffusion, Phys. A, 395 (2014), 228-235. doi: 10.1016/j.physa.2013.10.019
[30]	H. Q. Zhu, M. X. Wang, P. Y. Lai, General two-species interacting Lotka-Volterra system: Population dynamics and wave propagation, Phys. Rev. E, 97 (2018), 052413. doi: 10.1103/PhysRevE.97.052413
[31]	H. Q. Zhu, M. X. Wang, F. L. Hu, Interaction and coexistence with self-regulating species, Phys. A, 502 (2018), 447. doi: 10.1016/j.physa.2018.02.082
[32]	M. L. Rosenzweig, R. H. MacArthur, Graphical representation and stability conditions of predator-prey interactions, Am. Nat., 97 (1963), 209-223; doi: 10.1086/282272
[33]	J. P. Chen, H. D. Zhang, The qualitative analysis of two species predator-prey model with Holling's type Ⅲ functional response, Appl. Math. Mech., 7 (1986), 77-86. doi: 10.1007/BF01896254
[34]	C. S. Holling, The components of predation as revealed by a study of small mammal predation of the European pine sawfly, Can. Entomol., 91 (1959), 293-320. doi: 10.4039/Ent91293-5
[35]	C. S. Holling, Some characteristics of simple types of predation and parasitism, Can. Entomol., 91 (1959), 385-398. doi: 10.4039/Ent91385-7
[36]	C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, Mem. Entomol. Soc. Can., 45 (1959), 5-60.
[37]	R. Peng, J. Shi, M. Wang, Stationary pattern of a ratio-dependent food chain model with diffusion, SIAM J. Appl. Math., 67 (2007), 1479-1503. doi: 10.1137/05064624X
[38]	R. R. Vance, The Effect of Dispersal on Population Stability in One-Species, Discrete-Space Population Growth Models, Am. Nat., 123 (1984), 230. doi: 10.1086/284199
[39]	L. J. S. Allen, Persistence and extinction in Lotka-Volterra reaction-diffusion equations, Math. Biosci., 65 (1983), 1. doi: 10.1016/0025-5564(83)90068-8
[40]	Y. Takeuchi, Diffusion effect on stability of Lotka-Volterra models, Bull. Math. Biol., 48 (1986), 585-601. doi: 10.1016/S0092-8240(86)90009-1
[41]	A. Hastings, Global stability in Lotka-Volterra systems with diffusion, J. Math. Biol., 6 (1978) 163-168. doi: 10.1007/BF02450786
[42]	R. A. Fisher, The advance of advantageous genes, Ann. Eugenics, 7 (1937), 353.
[43]	A. Kolmogorov, I. Petrovskii, N. Piscounov, Selected Works of A. N. Kolmogorov, Vol. 2, I Math. Appl., 1991 (1991), 25.
[44]	M. M. Tang and P. C. Fife, Propagating fronts for competing species equations with diffusion, Arch. Ration. Mech. Anal., 73 (1980), 69. doi: 10.1007/BF00283257
[45]	J. I. Kanel, L. Zhou, Existence of wave front solutions and estimates of wave speed for a competition-diffusion system, Nonlinear Anal. TMA, 27 (1996), 579. doi: 10.1016/0362-546X(95)00221-G
[46]	E. Barbera, G. Consolo, G. Valenti, A two or three compartments hyperbolic reaction-diffusion model for the aquatic food chain, Math. Biosci. Eng., 12 (2015), 451. doi: 10.3934/mbe.2015.12.451
[47]	C. Erica, J. E. Paullet, J. P. Previte, Z. Walls, A Lotka- Volterra three-species food chain, Math. Mag., 75 (2002), 243-255. doi: 10.1080/0025570X.2002.11953139
[48]	O. De Feo, S. Rinaldi, Yield and dynamics of tri-trophic food chains, Am. Nat., 150 (1997), 328-345. doi: 10.1086/286068
[49]	C. Borrvall, B. Ebenman, T. Jonsson, Biodiversity lessens the risk of cascading extinction in model food webs, Ecol. Lett., 3 (2000), 131-136. doi: 10.1046/j.1461-0248.2000.00130.x
[50]	C. H. Chiu, S. B. Hsu, Extinction of top-predator in a three-level food-chain model, J. Math. Biol., 37 (1998), 372-380. doi: 10.1007/s002850050134
[51]	K. R. Crooks, M. E. Soulé, Mesopredator release and avifaunal extinctions in a fragmented system, Nature, 400 (1999), 563-566. doi: 10.1038/23028
[52]	K. S. McCann, J. B. Rasmussen, J. Umbanhowar, The dynamics of spatially coupled food webs, Eco. Lett., 8 (2005), 513-523. doi: 10.1111/j.1461-0248.2005.00742.x
[53]	M. X. Wang, Y. J. Li, P. Y. Lai, C. K. Chan, Model on cell movement, growth, differentiation and de-differentiation: Reaction-diffusion equation and wave propagation, Eur. Phys. J. E, 36 (2013), 65. doi: 10.1140/epje/i2013-13065-4
[54]	C, Liu, X. Fu, L. Liu, X. Ren, C. K. L. Chau, S. Li, et al., Sequential establishment of stripe patterns in an expanding cell population, Science, 334 (2011), 238. doi: 10.1126/science.1209042

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