Adaptation of species as response to climate change: Predator-prey mathematical model

Yadigar Sekerci; Yadigar Sekerci

doi:10.3934/math.2020251

AIMS Mathematics

2020, Volume 5, Issue 4: 3875-3898. doi: 10.3934/math.2020251

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Research article

Adaptation of species as response to climate change: Predator-prey mathematical model

Yadigar Sekerci ^,

Department of Mathematics, Arts and Science Faculty, Amasya University, 05189 Amasya, Turkey

Received: 27 December 2019 Accepted: 14 April 2020 Published: 23 April 2020
MSC : 93A30, 97M10, 97MXX

Most of the species currently threatened with extinction seem to be under the pressure of unsuitable environmental conditions; e.g., climate change, scarce food resource, habitat fragmentation. One should expect species to have forms of resilience against such extinction. The point here is to examine the effect of spatial gradients on species survival against increasing temperature arising from climate change. Therefore, we start with the question of whether, when faced with extinction stemming from climate change, a spatial gradient and a beachhead have the power to prevent extinction. This problem is addressed theoretically using a coupled reaction diffusion equation for a predator-prey system in which the prey experiences an Allee effect. It is demonstrated that there exists a relationship between the slope of the gradient and the beachhead at which the predator-prey system can stably survive. The tendency of the system can be defined by a function where the system includes the threshold point for extinction, that separates the areas of extinction and survival. The findings reveal that spatial gradient can be used as a precaution, when the species faces to extinction, for species to create new habitat and sustain its persistence. Therefore, in this paper, it is shown that, in theory, the recovery of species from unsuitable environmental conditions can be achieved. This can be possible by taking into account the spatial gradient to slow down the forthcoming ecological extinction, and thus extend the system a while as an adaptation mechanism.
- predator-prey system,
- spatial gradient,
- extinction,
- persistence,
- Allee effect,
- climate change
Citation: Yadigar Sekerci. Adaptation of species as response to climate change: Predator-prey mathematical model[J]. AIMS Mathematics, 2020, 5(4): 3875-3898. doi: 10.3934/math.2020251

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Abstract

Most of the species currently threatened with extinction seem to be under the pressure of unsuitable environmental conditions; e.g., climate change, scarce food resource, habitat fragmentation. One should expect species to have forms of resilience against such extinction. The point here is to examine the effect of spatial gradients on species survival against increasing temperature arising from climate change. Therefore, we start with the question of whether, when faced with extinction stemming from climate change, a spatial gradient and a beachhead have the power to prevent extinction. This problem is addressed theoretically using a coupled reaction diffusion equation for a predator-prey system in which the prey experiences an Allee effect. It is demonstrated that there exists a relationship between the slope of the gradient and the beachhead at which the predator-prey system can stably survive. The tendency of the system can be defined by a function where the system includes the threshold point for extinction, that separates the areas of extinction and survival. The findings reveal that spatial gradient can be used as a precaution, when the species faces to extinction, for species to create new habitat and sustain its persistence. Therefore, in this paper, it is shown that, in theory, the recovery of species from unsuitable environmental conditions can be achieved. This can be possible by taking into account the spatial gradient to slow down the forthcoming ecological extinction, and thus extend the system a while as an adaptation mechanism.

References

[1]	W. C. Allee, Animal Aggregations: A Study in General Sociology, The University of Chicago Press, Chicago, IL, USA, 1931.
[2]	A. Morozov, S. Petrovskii, B. L. Li, Spatiotemporal complexity of patchy invasion in a predatorprey system with the Allee effect, J. Theor. Biol., 238 (2006), 18-35. doi: 10.1016/j.jtbi.2005.05.021
[3]	S. Petrovskii, A. Morozov, E. Venturino, Allee effect makes possible patchy invasion in a predatorprey system, Ecol. Lett., 5 (2002), 345-352. doi: 10.1046/j.1461-0248.2002.00324.x
[4]	S. W. Yao, Z. P. Ma, Z. B. Cheng, Pattern formation of a diffusive predator-prey model with strong Allee effect and nonconstant death rate, Physica A, 527 (2019), 1-11.
[5]	R. Han, B. Dai, Spatiotemporal pattern formation and selection induced by nonlinear crossdiffusion in a toxic-phytoplankton-zooplankton model with Allee effect, Nonlinear Anal.: Real World Appl., 45 (2019), 822-853. doi: 10.1016/j.nonrwa.2018.05.018
[6]	S. Yan, D. Jia, T. Zhang, et al. Pattern dynamics in a diffusive predator-prey model with hunting cooperations, Chaos, Soliton. Fract., 130 (2020), 1-12.
[7]	S. Petrovskii, A. Morozov, B. L. Li, Regimes of biological invasion in a predator-prey system with the Allee effect, B. Math. Biol., 67 (2005), 637-661. doi: 10.1016/j.bulm.2004.09.003
[8]	H. Berestycki, L. Desvillettes, O. Diekmann, Can climate change lead to gap formation?, Ecol. Complex., 20 (2014), 264-270. doi: 10.1016/j.ecocom.2014.10.006
[9]	C. Cosner, Challenges in modelling biological invasions and population distributions in a changing climate, Ecol. Complex., 20 (2014), 258-263. doi: 10.1016/j.ecocom.2014.05.007
[10]	Y. Sekerci, Climate change effects on fractional order prey-predator model, Chaos, Soliton. Fract., 134 (2020), 1-16. doi: 10.1016/j.chaos.2020.109690
[11]	O. Bonnefon, J. Coville, J. Garnier, et al. The spatio-temporal dynamics of neutral genetic diversity, Ecol. Complex., 20 (2014), 282-292. doi: 10.1016/j.ecocom.2014.05.003
[12]	P. Kyriazopoulos, J. Nathan, E. Meron, Species coexistence by front pinning, Ecol. Complex., 20 (2014), 271-281. doi: 10.1016/j.ecocom.2014.05.001
[13]	J. T. Curtis, The Vegetation of Wisconsin: An Ordination of Plant Communities, University of Wisconsin Press, 1959.
[14]	R. H. Whittaker, Vegetation of the great smoky mountains, Ecol. Monogr., 26 (1956), 1-80. doi: 10.2307/1943577
[15]	M. Doebeli, U. Dieckmann, Speciation along environmental gradients, Nature, 421 (2003), 259-264. doi: 10.1038/nature01274
[16]	H. A. Gleason, The structure and development of the plant association, Bull. Torrey Bot. Club, 44 (1917), 463-81. doi: 10.2307/2479596
[17]	H. A. Gleason, The individualist concept of the plant association, Bull. Torrey Bot. Club, 53 (1926), 7-26. doi: 10.2307/2479933
[18]	H. A. Gleason, The individualistic concept of the plant association, Am. Midl. Nat., 21 (1939), 92-110. doi: 10.2307/2420377
[19]	M. Pascual, Diffusion-induced chaos in a spatial predator-prey system, Proc. R. Soc. Lond. B., 251 (1993), 1-7. doi: 10.1098/rspb.1993.0001
[20]	E. Post, Large-scale spatial gradients in herbivore population dynamics, Ecology, 86 (2005), 2320-2328. doi: 10.1890/04-0823
[21]	J. Z. Farkas, A. Y. Morozov, E. G. Arashkevich, et al. Revisiting the stability of spatially heterogeneous predator-prey systems under eutrophication, B. Math. Biol., 77 (2015), 1886-1908. doi: 10.1007/s11538-015-0108-2
[22]	L. Jonkers, H. Hillebrand, M. Kucera, Global change drives modern plankton communities away from the pre-industrial state, Nature, 570 (2019), 372-375. doi: 10.1038/s41586-019-1230-3
[23]	M. Siccha, M. Kucera, ForCenS, a curated database of planktonic foraminifera census counts in marine surface sediment samples, Sci. Data, 4 (2017), 1-12. doi: 10.1007/s40745-016-0096-6
[24]	G. T. Pecl, M. B. Araújo, J. D. Bell, et al. Biodiversity redistribution under climate change: Impacts on ecosystems and human well-being, Science, 355 (2017), 1-9.
[25]	S. Levin, L. A. Segel, Hypothesis for origin of planktonic patchiness, Nature, 259 (1976), 659. doi: 10.1038/259659a0
[26]	H. Malchow, S. V. Petrovskii, E. Venturino, Spatiotemporal Patterns in Ecology and Epidemiology: Theory, Models, and Simulation, CRC Press, 2007.
[27]	T. Singh, R. Dubey, V. N. Mishra, Spatial dynamics of predator-prey system with hunting cooperation in predators and type I functional response, AIMS Mathematics, 5 (2019), 673-684. doi: 10.3934/math.2020045
[28]	A. M. Turing, The chemical basis of morphogenesis, Philos. T. R. Soc. B., 237 (1952), 37-72.
[29]	S. A. Levin, T. M. Powell, J. H. Steele, Patch Dynamics, Springer Science & Business Media, 2012.
[30]	A. Morozov, K. Abbott, K. Cuddington, et al. Long transients in ecology: Theory and applications, Phys. Life Rev., 2019.
[31]	S. Petrovskii, Y. Sekerci, E. Venturino, Regime shifts and ecological catastrophes in a model of plankton-oxygen dynamics under the climate change, J. Theor. Biol., 424 (2017), 91-109. doi: 10.1016/j.jtbi.2017.04.018
[32]	Y. Sekerci, S. Petrovskii, Global warming can lead to depletion of oxygen by disrupting phytoplankton photosynthesis: a mathematical modelling approach, Geosciences, 8 (2018), 1-21. doi: 10.3390/geosciences8060201
[33]	Y. Sekerci, S. Petrovskii, Mathematical modelling of plankton-oxygen dynamics under the climate change, B. Math. Biol., 77 (2015), 2325-2353. doi: 10.1007/s11538-015-0126-0
[34]	A. S. Ackleh, L. J. Allen, J. Carter, Establishing a beachhead: a stochastic population model with an Allee effect applied to species invasion, Theor. Popul. Biol., 71 (2007), 290-300. doi: 10.1016/j.tpb.2006.12.006
[35]	Y. Sekerci, S. Petrovskii, Pattern formation in a model oxygen-plankton system, Computation, 6 (2018), 59. doi: 10.3390/computation6040059
[36]	J. Wang, J. Shi, J. Wei, Predator-prey system with strong Allee effect in prey, J. Math. Biol., 62 (2011), 291-331. doi: 10.1007/s00285-010-0332-1
[37]	J. Wang, J. Shi, J. Wei, Nonexistence of periodic orbits for predator-prey system with strong Allee effect in prey populations, Electron. J. Differ. Eq., 2013 (2013), 1-14. doi: 10.1186/1687-1847-2013-1
[38]	G. A. Van Voorn, L. Hemerik, M. P. Boer, et al., Heteroclinic orbits indicate overexploitation in predator-prey systems with a strong Allee effect, Math. Biosci., 209 (2007), 451-469. doi: 10.1016/j.mbs.2007.02.006
[39]	R. M. Nisbet, W. Gurney, Modelling Fluctuating Populations, Chichester, 1982.
[40]	J. D. Murray, Mathematical Biology, Springer, Berlin, 1993.
[41]	J. A. Sherratt, Periodic travelling waves in cyclic predator-prey systems, Ecol. Lett., 4 (2001), 30-37. doi: 10.1046/j.1461-0248.2001.00193.x
[42]	M. H. Wang, M. Kot, Speeds of invasion in a model with strong or weak Allee effects, Math. Biosci., 171 (2001), 83-97. doi: 10.1016/S0025-5564(01)00048-7
[43]	S. Chen, H. Yang, J. Wei, Global dynamics of two phytoplankton-zooplankton models with toxic substances effect, J. Appl. Anal. Comput., 9 (2019), 796-809.
[44]	H. Liu, T. Li, F. Zhang, A prey-predator model with Holling II functional response and the carrying capacity of predator depending on its prey, J. Appl. Anal. Comput., 8 (2018), 1464-1474.
[45]	M. A. Lewis, P. Kareiva, Allee dynamics and the spread of invading organisms, Theor. Popul. Biol., 43 (1993), 141-158. doi: 10.1006/tpbi.1993.1007
[46]	S. V. Petrovskii, H. Malchow, Wave of chaos: New mechanism of pattern formation in spatiotemporal population dynamics, Theor. Popul. Biol., 59 (2001), 157-174. doi: 10.1006/tpbi.2000.1509
[47]	W. F. Fagan, J. G. Bishop, Trophic interactions during primary succession: Herbivores slow a plant reinvasion at Mount St. Helens, Am. Nat., 155 (2000), 238-251. doi: 10.1086/303320
[48]	M. R. Owen, M. A. Lewis, How predation can slow, stop or reverse a prey invasion, B. Math. Biol., 63 (2001), 655-684. doi: 10.1006/bulm.2001.0239
[49]	S. Petrovskii, H. Malchow, B. L. Li, An exact solution of a diffusive predator-prey system, Proc. R. Soc. A., 461 (2005), 1029-1053. doi: 10.1098/rspa.2004.1404
[50]	S. Petrovskii, A. Morozov, M. Sieber Noise can prevent onset of chaos in spatiotemporal population dynamics, Eur. Phys. J. B., 78 (2010), 253-264.
[51]	W. Alharbi, S. Petrovskii, Critical domain problem for the reaction-telegraph equation model of population dynamics, Mathematics, 6 (2018), 1-15. doi: 10.3390/math6040059
[52]	J. Martin, B. van Moorter, E. Revilla, et al. Reciprocal modulation of internal and external factors determines individual movements, J. Anim. Ecol., 82 (2013), 290-300. doi: 10.1111/j.1365-2656.2012.02038.x
[53]	B. Chakraborty, N. Bairagi, Complexity in a prey-predator model with prey refuge and diffusion, Ecol. Complex., 37 (2019), 11-23. doi: 10.1016/j.ecocom.2018.10.004
[54]	V. Dakos, E. H. van Nes, R. Donangelo, et al. Spatial correlation as leading indicator of catastrophic shifts, Theor. Ecol., 3 (2010), 163-174. doi: 10.1007/s12080-009-0060-6
[55]	A. Hastings, K. C. Abbott, K. Cuddington, et al. Transient phenomena in ecology, Science, 361 (2018), 1-11. doi: 10.1126/science.aat6412
[56]	S. Kéfi, M. Rietkerk, C. L. Alados, et al. Spatial vegetation patterns and imminent desertification in Mediterranean arid ecosystems, Nature, 449 (2007), 213-217. doi: 10.1038/nature06111
[57]	M. Pascual, F. Guichard, Critically and disturbance in spatial ecological system, Trends Ecol. Evol., 20 (2005), 88-95. doi: 10.1016/j.tree.2004.11.012
[58]	M. Rietkerk, S. C. Dekker, P. C. de Ruiter, et al. Self-organized patchiness and catastrophic shifts in ecosystems, Science, 305 (2004), 1926-1929. doi: 10.1126/science.1101867

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