Local dynamics and coexistence of predator–prey model with directional dispersal of predator

Kwangjoong Kim; Wonhyung Choi; Kwangjoong Kim; Wonhyung Choi

doi:10.3934/mbe.2020351

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 6: 6737-6755. doi: 10.3934/mbe.2020351

Previous Article Next Article

Research article Special Issues

Local dynamics and coexistence of predator–prey model with directional dispersal of predator

Kwangjoong Kim ¹,
Wonhyung Choi ^{2,
,
,}

1.
College of General Education, Kookmin University, 77, Jeongneung-ro, Seoul, 02707, Korea
2.
Industry-Academic Cooperation Foundation, Kookmin University, 77, Jeongneung-ro, Seoul, 02707, Korea
² The author is now with the Department of Mathematics, Tamkang University, 151, Yingzhuan Road, Tamsui, New Taipei City, 251301, Taiwan.

Received: 15 June 2020 Accepted: 23 September 2020 Published: 30 September 2020

In this paper, we study the effect of directional dispersal of a predator on a predator– prey model. The prey is assumed to have traits making it undetectable to the predator and difficult to chase the prey directly. Directional dispersal of the predator is described when the predator has learned the high hunting efficiency in certain areas, thereby dispersing toward these areas instead of directly chasing the prey. We investigate the stability of the semi-trivial solution and the existence of a coexistence steady-state. Moreover, we show that the predator that moves toward a high-predation area may make the predators survive under the condition the predators cannot survive when they disperse randomly. The results are obtained through eigenvalue analysis and fixed-point index theory. Finally, we present the numerical simulation and its biological interpretations based on the obtained results.
- predator–prey model,
- directional dispersal,
- ratio-dependent functional responses,
- local stability/instability,
- coexistence steady-state
Citation: Kwangjoong Kim, Wonhyung Choi. Local dynamics and coexistence of predator–prey model with directional dispersal of predator[J]. Mathematical Biosciences and Engineering, 2020, 17(6): 6737-6755. doi: 10.3934/mbe.2020351

Related Papers:

Abstract

In this paper, we study the effect of directional dispersal of a predator on a predator– prey model. The prey is assumed to have traits making it undetectable to the predator and difficult to chase the prey directly. Directional dispersal of the predator is described when the predator has learned the high hunting efficiency in certain areas, thereby dispersing toward these areas instead of directly chasing the prey. We investigate the stability of the semi-trivial solution and the existence of a coexistence steady-state. Moreover, we show that the predator that moves toward a high-predation area may make the predators survive under the condition the predators cannot survive when they disperse randomly. The results are obtained through eigenvalue analysis and fixed-point index theory. Finally, we present the numerical simulation and its biological interpretations based on the obtained results.

References

[1]	M. Iida, M. Mimura, H. Ninomiya, Diffusion, cross-diffusion and competitive interaction, J. Math. Biol., 53 (2006), 617-641.
[2]	W. Ko, K. Ryu, On a predator-prey system with cross diffusion representing the tendency of predators in the presence of prey species, J. Math. Anal. Appl., 341 (2008), 1133-1142. doi: 10.1016/j.jmaa.2007.11.018
[3]	T. Kadota, K. Kuto, Positive steady states for a prey-predator model with some nonlinear diffusion terms, J. Math. Anal. Appl., 323 (2006), 1387-1401. doi: 10.1016/j.jmaa.2005.11.065
[4]	K. Kuto, A strongly coupled diffusion effect on the stationary solution set of a prey-predator model, Adv. Differential. Equ., 12 (2007), 145-172.
[5]	K. Kuto, Y. Yamada, Coexistence problem for a prey-predator model with density-dependent diffusion, Nonlinear Anal.-Theor., 71 (2009), e2223-e2232.
[6]	K. Kuto, Y. Yamada, Limiting characterization of stationary solutions for a prey-predator model with nonlinear diffusion of fractional type, Differ. Integral. Equ., 22 (2009), 725-752.
[7]	Y. Lou, W. Ni, Diffusion vs cross-diffusion: an elliptic approach, J. Differ. Equations, 154 (1999), 157-190. doi: 10.1006/jdeq.1998.3559
[8]	K. Ryu, I. Ahn, Positive steady-states for two interacting species models with linear self-cross diffusions, Discrete Contin. Dyn-A, 9 (2003), 1049. doi: 10.3934/dcds.2003.9.1049
[9]	K. Ryu, I. Ahn, Coexistence theorem of steady states for nonlinear self-cross diffusion systems with competitive dynamics, J. Math. Anal. Appl., 283 (2003), 46-65. doi: 10.1016/S0022-247X(03)00162-8
[10]	N. Shigesada, K. Kawasaki, E. Teramoto, Spatial segregation of interacting species, J. Theor. Biol., 79 (1979), 83-99. doi: 10.1016/0022-5193(79)90258-3
[11]	I. Averill, K. Lam, Y. Lou, The role of advection in a two-species competition model: a bifurcation approach, volume 245. American Mathematical Society, 2017.
[12]	X. Chen, K. Lam, Y. Lou, Dynamics of a reaction-diffusion-advection model for two competing species, Discrete Contin. Dyn. S., 32 (2012), 3841-3859. doi: 10.3934/dcds.2012.32.3841
[13]	C. Cosner, Reaction-diffusion-advection models for the effects and evolution of dispersal, Discrete Contin. Dyn-A, 34 (2014), 1701-1745. doi: 10.3934/dcds.2014.34.1701
[14]	R. S. Cantrell, C. Cosner, Y. Lou, Movement toward better environments and the evolution of rapid diffusion, Math. Biosci., 204 (2006), 199-214. doi: 10.1016/j.mbs.2006.09.003
[15]	R. S. Cantrell, C. Cosner, Y. Lou, Advection-mediated coexistence of competing species, P. Roy. Soc. Edinb. A., 137 (2007), 497-518. doi: 10.1017/S0308210506000047
[16]	R. S. Cantrell, C. Cosner, Y. Lou, Approximating the ideal free distribution via reaction-diffusion- advection equations, J. Differ. Equations, 245 (2008), 3687-3703. doi: 10.1016/j.jde.2008.07.024
[17]	C. Cosner, Y. Lou, Does movement toward better environments always benefit a population?, J. Math. Anal. Appl., 277 (2003), 489-503. doi: 10.1016/S0022-247X(02)00575-9
[18]	K. Kuto, T. Tsujikawa, Limiting structure of steady-states to the lotka-volterra competition model with large diffusion and advection, J. Differ. Equations, 258 (2015), 1801-1858. doi: 10.1016/j.jde.2014.11.016
[19]	K.-Y. Lam, W.-M. Ni, Advection-mediated competition in general environments, J. Differ. Equations, 257 (2014), 3466-3500. doi: 10.1016/j.jde.2014.06.019
[20]	K.-Y. Lam, W. M. Ni, Limiting profiles of semilinear elliptic equations with large advection in population dynamics, Discrete Contin. Dyn. Syst., 28 (2010), 1051-1067. doi: 10.3934/dcds.2010.28.1051
[21]	E. Cho, Y.-J. Kim, Starvation driven diffusion as a survival strategy of biological organisms, B. Math. Biol., 75 (2013), 845-870. doi: 10.1007/s11538-013-9838-1
[22]	W. Choi, S. Baek, I. Ahn, Intraguild predation with evolutionary dispersal in a spatially heterogeneous environment, J. Math. Biol., 78 (2019), 2141-2169. doi: 10.1007/s00285-019-01336-5
[23]	W. Choi, I. Ahn, Strong competition model with non-uniform dispersal in a heterogeneous environment, Appl. Math. Lett., 88 (2019), 96-102. doi: 10.1016/j.aml.2018.08.014
[24]	W. Choi, I. Ahn, Non-uniform dispersal of logistic population models with free boundaries in a spatially heterogeneous environment, J. Math. Anal. Appl., 479 (2019), 283-314. doi: 10.1016/j.jmaa.2019.06.027
[25]	W. Choi, I. Ahn, Predator-prey interaction systems with non-uniform dispersal in a spatially heterogeneous environment, J. Math. Anal. Appl., 485 (2020), 123860. doi: 10.1016/j.jmaa.2020.123860
[26]	Y.-J. Kim, O. Kwon, F. Li, Evolution of dispersal toward fitness, B. Math. Biol., 75 (2013), 2474- 2498. doi: 10.1007/s11538-013-9904-8
[27]	Y.-J. Kim, O. Kwon, F. Li, Global asymptotic stability and the ideal free distribution in a starvation driven diffusion, J. Math. Biol., 68 (2014), 1341-1370. doi: 10.1007/s00285-013-0674-6
[28]	Y.-J. Kim, O. Kwon, Evolution of dispersal with starvation measure and coexistence, B. Math. Biol., 78 (2016), 254-279. doi: 10.1007/s11538-016-0142-8
[29]	W. Choi, I. Ahn, Effect of prey-taxis on predator's invasion in a spatially heterogeneous environment, Appl. Math. Lett., 98 (2019), 256-262. doi: 10.1016/j.aml.2019.06.021
[30]	S. Wu, J. Shi, B. Wu, Global existence of solutions and uniform persistence of a diffusive predator- prey model with prey-taxis, J. Differ. Equations, 260 (2016), 5847-5874. doi: 10.1016/j.jde.2015.12.024
[31]	H. Jin, Z. Wang, Global stability of prey-taxis systems, J. Differ. Equations, 262 (2017), 1257- 1290. doi: 10.1016/j.jde.2016.10.010
[32]	Y. Tao, Global existence of classical solutions to a predator-prey model with nonlinear prey-taxis, Nonlinear Anal.-Real., 11 (2010), 2056-2064. doi: 10.1016/j.nonrwa.2009.05.005
[33]	X. He, S. Zheng, Global boundedness of solutions in a reaction-diffusion system of predator-prey model with prey-taxis, Appl. Math. Lett., 49 (2015), 73-77. doi: 10.1016/j.aml.2015.04.017
[34]	C. Li, X. Wang, Y. Shao, Steady states of a predator-prey model with prey-taxis, Nonlinear Anal.- Theor., 97 (2014), 155-168. doi: 10.1016/j.na.2013.11.022
[35]	P. A. Abrams, L. R. Ginzburg, The nature of predation: prey dependent, ratio dependent or neither?, Trends Ecol. Evol., 15 (2000), 337-341.
[36]	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
[37]	C. Cosner, D. L. DeAngelis, J. S. Ault, D. B. Olson, Effects of spatial grouping on the functional response of predators, Theor. Popul. Biol., 56 (1999), 65-75. doi: 10.1006/tpbi.1999.1414
[38]	J. M. Culp, N. E. Glozier, G. J. Scrimgeour, Reduction of predation risk under the cover of darkness: avoidance responses of mayfly larvae to a benthic fish, Oecologia, 86 (1991), 163-169. doi: 10.1007/BF00317527
[39]	F. Mougeot, V. Bretagnolle, Predation risk and moonlight avoidance in nocturnal seabirds, J. Avian. Biol., 31 (2000), 376-386. doi: 10.1034/j.1600-048X.2000.310314.x
[40]	T. Caro, Antipredator defenses in birds and mammals, University of Chicago Press, 2005.
[41]	H. B. Cott, Adaptive coloration in animals, 1940.
[42]	J. M. Hemmi, Predator avoidance in fiddler crabs: 1. escape decisions in relation to the risk of predation, Anim. Behav., 69 (2005), 603-614. doi: 10.1016/j.anbehav.2004.06.018
[43]	W. J. Bell, Searching behaviour: the behavioural ecology of finding resources, Springer Science & Business Media, 2012.
[44]	S. Benhamou, Spatial memory and searching efficiency, Anim. Behav., 47 (1994), 1423-1433. doi: 10.1006/anbe.1994.1189
[45]	S. Benhamou, Bicoordinate navigation based on non-orthogonal gradient fields, J. Theo. Biol., 225 (2003), 235-239. doi: 10.1016/S0022-5193(03)00242-X
[46]	W. F. Fagan, M. A. Lewis, M. Auger-Meth ′ e, T. Avgar, S. Benhamou, G. Breed, et al., Spatial ′ memory and animal movement, Ecol. Lett., 16 (2013), 1316-1329.
[47]	S. M. Flaxman, Y. Lou, Tracking prey or tracking the prey's resource? mechanisms of movement and optimal habitat selection by predators, J. Theor. Biol., 256 (2009), 187-200. doi: 10.1016/j.jtbi.2008.09.024
[48]	S. M. Flaxman, Y. Lou, F. G. Meyer, Evolutionary ecology of movement by predators and prey, Theor. Ecol., 4 (2011), 255-267. doi: 10.1007/s12080-011-0120-6
[49]	A. M. Kittle, M. Anderson, T. Avgar, J. A. Baker, G. S. Brown, J. Hagens, et al., Landscape-level wolf space use is correlated with prey abundance, ease of mobility, and the distribution of prey habitat, Ecosphere, 8 (2017), e01783.
[50]	H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, In Function spaces, differential operators and nonlinear analysis, pages 9-126. Springer, 1993.
[51]	R. S. Cantrell, C. Cosner, Spatial ecology via reaction-diffusion equations, John Wiley & Sons, 2004.
[52]	L. J. S. Allen, B. M. Bolker, Y. Lou, A. L. Nevai, Asymptotic profiles of the steady states for an sis epidemic reaction-diffusion model, Discrete Contin. Dyn. Syst. Ser. A, 21 (2008), 1. doi: 10.3934/dcds.2008.21.1
[53]	X. He, W.-M. Ni, Global dynamics of the lotka-volterra competition-diffusion system: Diffusion and spatial heterogeneity i, Commun. Pur. Appl. Math., 69 (2016), 981-1014. doi: 10.1002/cpa.21596
[54]	E. N. Dancer, On the indices of fixed points of mappings in cones and applications, J. Math. Anal. Appl., 91 (1983), 131-151. doi: 10.1016/0022-247X(83)90098-7
[55]	L. Li, Coexistence theorems of steady states for predator-prey interacting systems, T. Am. Math. Soc., 305 (1988), 143-166. doi: 10.1090/S0002-9947-1988-0920151-1
[56]	M. Wang, Z. Li, Q. Ye, Existence of positive solutions for semilinear elliptic system, In Qualitative aspects and applications of nonlinear evolution equations, 1991.
[57]	K. Ryu, I. Ahn, Positive solutions for ratio-dependent predator-prey interaction systems, J. Differ. Equations, 218 (2005), 117-135. doi: 10.1016/j.jde.2005.06.020

Reader Comments

Your name:*

Email:*
© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)