Abstract
This study considers a situation in which a predator can change its dispersal rate according to its satisfaction with foraging prey in a predator–prey interaction. However, since it is impossible to accurately determine the magnitude of the density of prey that is favorable to a predator’s survival in an area, the predator determines the movement rate through inaccurate judgment. In this situation, we investigate the effect of the predator’s decision about its movement on fitness. To achieve our goal, we consider a predator–prey model with nonuniform predator dispersal, called prey-induced dispersal (PYID), in which the spread of predators is small when the prey density is larger than a certain value, and when the prey density is smaller than a particular value, a large spread of predators occurs. To understand how PYID affects the dynamics and coexistence of the system in a spatially heterogeneous region, we examine a model with Holling-type II functional responses under no-flux boundary conditions wherein the predators move according to the PYID. We study the local stability of the semitrivial solution of models with PYID and linear dispersal where the predator is absent. Furthermore, we investigate the local/global bifurcation from the semitrivial solution of models with two different dispersals. We conclude that in most cases, nonuniform dispersal of predators following PYID promotes predator fitness; however, there is a case in which PYID does not increase predator fitness. If a predator’s satisfaction degree regarding the prey density is higher than a certain level, there may exist a case that is not beneficial for predators in terms of their fitness. However, if the satisfaction level of predators regarding prey density is relatively low, predators following PYID will take advantage of fitness. More precisely, if predators are dissatisfied with the amount of prey in a region and move quickly, even for abundant prey density, they may not benefit from PYID. Meanwhile, if predators change their motility when they are appropriately satisfied with the amount of prey, they will obtain a survival advantage. We obtain the results by analyzing an eigenvalue problem at the semitrivial solution from the linearized operators derived from the models.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abrahams MV (1986) Patch choice under perceptual constraints: a cause for departures from an ideal free distribution. Behav Ecol Sociobiol 19(6):409–415. https://doi.org/10.1007/BF00300543
Ahn I, Yoon C (2019) Global well-posedness and stability analysis of prey-predator model with indirect prey- taxis. J Differ Equ 268(8):4222–4255. https://doi.org/10.1016/j.jde.2019.10.019
Ahn I, Yoon C (2021) Global solvability of prey–predator models with indirect predator-taxis. Z Angew Math Phys 72(1):1–20. https://doi.org/10.1007/s00033-020-01461-y
Amano T, Katayama N (2009) Hierarchical movement decisions in predators: effects of foraging experience at more than one spatial and temporal scale. Ecology 90(12):3536–3545. https://doi.org/10.1890/08-1910.1
Averill I, Lam KY, Lou Y (2017) The role of advection in a two-species competition model: a bifurcation approach, vol 245, no 1161. Mem. Amer. Math. Soc., Providence
Bailey H, Thompson P (2006) Quantitative analysis of bottlenose dolphin movement patterns and their relationship with foraging. J Anim Ecol 75(2):456–465. https://doi.org/10.1111/j.1365-2656.2006.01066.x
Bailey H, Lyubchich V, Wingfield J, Fandel A, Garrod A, Rice AN (2019) Empirical evidence that large marine predator foraging behavior is consistent with area-restricted search theory. Ecology 100(8):e02743. https://doi.org/10.1002/ecy.2743C
Banks CJ (1957) The behaviour of individual coccinellid larvae on plants. Anim Behav 5(1):12–24. https://doi.org/10.1016/S0950-5601(57)80039-2
Berec L (2000) Mixed encounters, limited perception and optimal foraging. Bull Math Biol 62(5):849–868. https://doi.org/10.1006/bulm.2000.0179
Braverman E, Braverman L (2009) Optimal harvesting of diffusive models in a nonhomogeneous environment. Nonlinear Anal 71(12):e2173–e2181. https://doi.org/10.1016/j.na.2009.04.025
Braverman E, Kamrujjaman M, Korobenko L (2015) Competitive spatially distributed population dynamics models: does diversity in diffusion strategies promote coexistence? Math Biosci 264:63–73. https://doi.org/10.1016/j.mbs.2015.03.004
Cantrell RS, Cosner C (2004) Spatial ecology via reaction–diffusion equations. Wiley, Hoboken
Cantrell RS, Cosner C, Lou Y (2006) Movement toward better environments and the evolution of rapid diffusion. Math Biosci 204(2):199–214. https://doi.org/10.1016/j.mbs.2006.09.003
Cantrell RS, Cosner C, Lou Y (2007) Advection-mediated coexistence of competing species. Proc R Soc Edinb Sect A Math 137(3):497–518. https://doi.org/10.1017/S030821050600004
Cantrell RS, Cosner C, Lou Y (2010) Evolution of dispersal and the ideal free distribution. Math Biosci Eng 7(1):17. https://doi.org/10.3934/mbe.2010.7.17
Charnov EL (1976) Optimal foraging, the marginal value theorem. Theor Popul Biol 9(2):129–136. https://doi.org/10.1016/0040-5809(76)90040-X
Chen X, Lam KY, Lou Y (2012) Dynamics of a reaction–diffusion–advection model for two competing species. Discrete Contin Dyn Syst 32(11):3841. https://doi.org/10.3934/dcds.2012.32.3841
Cho E, Kim YJ (2013) Starvation driven diffusion as a survival strategy of biological organisms. Bull Math Biol 75(5):845–870. https://doi.org/10.1007/s11538-013-9838-1
Choi W, Ahn I (2019a) Effect of prey-taxis on predator’s invasion in a spatially heterogeneous environment. Appl Math Lett 98:256–262. https://doi.org/10.1016/j.aml.2019.06.021
Choi W, Ahn I (2019b) Strong competition model with non-uniform dispersal in a heterogeneous environment. Appl Math Lett 88:96–102. https://doi.org/10.1016/j.aml.2018.08.014
Choi W, Baek S, Ahn I (2019) Intraguild predation with evolutionary dispersal in a spatially heterogeneous environment. J Math Biol 78(7):2141–2169. https://doi.org/10.1007/s00285-019-01336-5
Choi W, Ahn I (2020) Predator–prey interaction systems with non-uniform dispersal in a spatially heterogeneous environment. J Math Anal Appl 485(2):123860. https://doi.org/10.1016/j.jmaa.2020.123860
Choi W, Ahn I (2022) Predator invasion in predator–prey model with prey-taxis in spatially heterogeneous environment. Nonlinear Anal Real World Appl 65:103495. https://doi.org/10.1016/j.nonrwa.2021.103495
Conway JB (2019) A course in functional analysis, vol 96. Springer, Berlin
Cosner C (2014) Reaction–diffusion–advection models for the effects and evolution of dispersal. Discrete Contin Dyn Syst 34(5):1701. https://doi.org/10.3934/dcds.2014.34.1701
Cosner C, Lou Y (2003) Does movement toward better environments always benefit a population? J Math Anal Appl 277(2):489–503. https://doi.org/10.1016/S0022-247X(02)00575-9
Crandall MG, Rabinowitz PH (1971) Bifurcation from simple eigenvalues. J Funct Anal 8(2):321–340. https://doi.org/10.1016/0022-1236(71)90015-2
Crandall MG, Rabinowitz PH (1973) Bifurcation, perturbation of simple eigenvalues and linearized stability. Arch Ration Mech Anal 52:161–180
Curio E (2012) The ethology of predation, vol 7. Springer, Berlin
Dieckman U, O’Hara B, Weisser W (1999) The evolutionary ecology of dispersal. Trends Ecol Evol 14(3):88–90. https://doi.org/10.1016/S0169-5347(98)01571-7
Dockery J, Hutson V, Mischaikow K, Pernarowski M (1998) The evolution of slow dispersal rates: a reaction diffusion model. J Math Biol 37(1):61–83. https://doi.org/10.1007/s002850050120
Evans HF (1976) The searching behaviour of Anthocoris confusus (Reuter) in relation to prey density and plant surface topography. Ecol Entomol 1(3):163–169. https://doi.org/10.1111/j.1365-2311.1976.tb01219.x
Hamer KC, Humphreys EM, Magalhaes MC, Garthe S, Hennicke J, Peters G, Gremillet D, Skov H, Wanless S (2009) Fine-scale foraging behaviour of a medium-ranging marine predator. J Anim Ecol 78(4):880–889. https://doi.org/10.1111/j.1365-2656.2009.01549.x
Hassell MP (1978) The dynamics of arthropod predator–prey systems. Princeton University Press, Princeton
Hauzy C, Gauduchon M, Hulot FD, Loreau M (2010) Density-dependent dispersal and relative dispersal affect the stability of predator–prey metacommunities. J Theoret Biol 266(3):458–469. https://doi.org/10.1016/j.jtbi.2010.07.008
He X, Zheng S (2015) Global boundedness of solutions in a reaction–diffusion system of predator–prey model with prey-taxis. Appl Math Lett 49:73–77. https://doi.org/10.1016/j.aml.2015.04.017
Jin HY, Wang ZA (2017) Global stability of prey-taxis systems. J Differ Equ 262(3):1257–1290. https://doi.org/10.1016/j.jde.2016.10.010
Johnson ML, Gaines MS (1990) Evolution of dispersal: theoretical models and empirical tests using birds and mammals. Annu Rev Ecol Syst 21(1):449–480. https://doi.org/10.1146/annurev.es.21.110190.002313
Kareiva P, Odell G (1987) Swarms of predators exhibit “preytaxis’’ if individual predators use area-restricted search. Am Nat 130(2):233–270. https://doi.org/10.1086/284707
Keeling, M (1999) Spatial models of interacting populations. In: McGlade J (ed) Advanced ecological theory: principles and applications. Blackwell Science Ltd., Oxford.
Kim YJ, Kwon O (2016) Evolution of dispersal with starvation measure and coexistence. Bull Math Biol 78(2):254–279. https://doi.org/10.1007/s11538-016-0142-8
Kim YJ, Kwon O, Li F (2013) Evolution of dispersal toward fitness. Bull Math Biol 75(12):2474–2498. https://doi.org/10.1007/s11538-013-9904-8
Kim YJ, Kwon O, Li F (2014) Global asymptotic stability and the ideal free distribution in a starvation driven diffusion. J Math Biol 68(6):1341–1370. https://doi.org/10.1007/s00285-013-0674-6
Korobenko L, Braverman E (2009) A logistic model with a carrying capacity driven diffusion. Can Appl Math Q 17(1):85–104
Korobenko L, Braverman E (2012) On logistic models with a carrying capacity dependent diffusion: stability of equilibria and coexistence with a regularly diffusing population. Nonlinear Anal Real World Appl 13(6):2648–2658. https://doi.org/10.1016/j.nonrwa.2011.12.027
Korobenko L, Braverman E (2014) On evolutionary stability of carrying capacity driven dispersal in competition with regularly diffusing populations. J Math Biol 69(5):1181–1206. https://doi.org/10.1007/s00285-013-0729-8
Kuto K, Yamada Y (2012) On limit systems for some population models with cross-diffusion. Discrete Contin Dyn Syst Ser B 17(8):2745. https://doi.org/10.3934/dcdsb.2012.17.2745
Lam KY (2011) Concentration phenomena of a semilinear elliptic equation with large advection in an ecological model. J Differ Equ 250(1):161–181. https://doi.org/10.1016/j.jde.2010.08.028
Lam KY, Ni WM (2010) Limiting profiles of semilinear elliptic equations with large advection in population dynamics. Discrete Contin Dyn Syst 28(3):1051. https://doi.org/10.3934/dcds.2010.28.1051
Lam KY, Ni WM (2014) Advection-mediated competition in general environments. J Differ Equ 1:1. https://doi.org/10.1016/j.jde.2014.06.019
López-Gómez J (2001) Spectral theory and nonlinear functional analysis. CRC Press, Boca Raton
López-Gómez J (2016) Global bifurcation for Fredholm operators. Rend Istit Mat Univ Trieste 48:539–564. https://doi.org/10.13137/2464-8728/13172
Li C, Wang X, Shao Y (2014) Steady states of a predator–prey model with prey-taxis. Nonlinear Anal 97:155–168. https://doi.org/10.1016/j.na.2013.11.022
Lou Y, Ni WM (1996) Diffusion, self-diffusion and cross-diffusion. J Differ Equ 131(1):79–131. https://doi.org/10.1006/jdeq.1996.0157
Lou Y, Wang B (2017) Local dynamics of a diffusive predator–prey model in spatially heterogeneous environment. J Fixed Point Theory Appl 19(1):755–772. https://doi.org/10.1007/s11784-016-0372-2
Lou Y, Ni WM, Yotsutani S (2004) On a limiting system in the Lotka–Volterra competition with cross-diffusion. Discrete Contin Dyn Syst 10(1 &2):435. https://doi.org/10.3934/dcds.2004.10.435
Lou Y, Tao Y, Winkler M (2017) Nonexistence of nonconstant steady-state solutions in a triangular cross-diffusion model. J Differ Equ 262(10):5160–5178. https://doi.org/10.1016/j.jde.2017.01.017
McPeek MA, Holt RD (1992) The evolution of dispersal in spatially and temporally varying environments. Am Nat 140(6):1010–1027. https://doi.org/10.1086/285453
Murdie G, Hassell MP (1973) Food distribution, searching success and predator–prey models. In: The mathematical theory of the dynamics of biological populations, pp 87–101. Academic Press, London, UK.
Okubo A, Levin SA (2001) Diffusion and ecological problems: modern perspectives, vol 14. Springer, New York
Rabinowitz PH (1971) Some global results for nonlinear eigenvalue problems. J Funct Anal 7(3):487–513. https://doi.org/10.1016/0022-1236(71)90030-9
Ramanantoanina A, Hui C, Ouhinou A (2011) Effects of density-dependent dispersal behaviours on the speed and spatial patterns of range expansion in predator–prey metapopulations. Ecol Model 222(19):3524–3530. https://doi.org/10.1016/j.ecolmodel.2011.08.010
Shi J, Wang X (2009) On global bifurcation for quasilinear elliptic systems on bounded domains. J Differ Equ 246(7):2788–2812. https://doi.org/10.1016/j.jde.2008.09.009
Skellam JG (1972) Some philosophical aspects of mathematical modelling in empirical science with special reference to ecology. Math Models Ecol 13(8):13–28.
Smith JN (1971) Studies of the searching behaviour and prey recognition of certain vertebrate predators. Doctoral dissertation, University of Oxford
Smoller J (2012) Shock waves and reaction–diffusion equations, vol 258. Springer, Berlin
Thums M, Bradshaw CJ, Hindell MA (2011) In situ measures of foraging success and prey encounter reveal marine habitat-dependent search strategies. Ecology 92(6):1258–1270. https://doi.org/10.1890/09-1299.1
Williamson CE (1981) Foraging behavior of a freshwater copepod: frequency changes in looping behavior at high and low prey densities. Oecologia 50(3):332–336. https://doi.org/10.1007/BF00344972
Wu S, Shi J, Wu B (2016) Global existence of solutions and uniform persistence of a diffusive predator–prey model with prey-taxis. J Differ Equ 260(7):5847–5874. https://doi.org/10.1016/j.jde.2015.12.024
Wu S, Wang J, Shi J (2018) Dynamics and pattern formation of a diffusive predator–prey model with predator-taxis. Math Models Methods Appl Sci 28(11):2275–2312. https://doi.org/10.1142/S0218202518400158
Yoon C (2021) Global dynamics of a Lotka–Volterra type prey–predator model with diffusion and predator-taxis. Appl Anal. https://doi.org/10.1080/00036811.2021.1898598
Zach R, Falls JB (1977) Influence of capturing a prey on subsequent search in the ovenbird (Aves: Parulidae). Can J Zool 55(12):1958–1969. https://doi.org/10.1139/z77-253
Acknowledgements
The authors greatly appreciate the valuable comments and suggestions of anonymous reviewers of this paper which significantly improved the original version. In particular, the authors would like to thank the reviewers for suggesting many references that supported the ecological link for the model considered in the paper. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (NRF-2020R1F1A1A01054935, and NRF-2019K2A9A2A06025237) and Ministry of Education (NRF-2021R1A6A3A01086879).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Chang, Y., Choi, W. & Ahn, I. On the Fitness of Predators with Prey-Induced Dispersal in a Habitat with Spatial Heterogeneity. Bull Math Biol 84, 111 (2022). https://doi.org/10.1007/s11538-022-01069-5
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11538-022-01069-5