# American Institute of Mathematical Sciences

August  2015, 20(6): 1591-1607. doi: 10.3934/dcdsb.2015.20.1591

## Spatial population dynamics in a producer-scrounger model

 1 Department of Mathematics, University of Miami, Coral Gables, FL 33146, United States 2 Department of Mathematics, University of Central Florida, Orlando, FL 32816, United States

Received  November 2013 Revised  January 2014 Published  June 2015

The spatial population dynamics of an ecological system involving producers and scroungers is studied using a reaction-diffusion model. The two populations move randomly and increase logistically, with birth rates determined by the amount of resource acquired. Producers can obtain the resource directly from the environment, but must surrender a proportion of their discoveries to nearby scroungers through a process known as scramble kleptoparasitism. The proportion of resources stolen by a scrounger from nearby producers decreases as the local scrounger density increases. Parameter combinations which allow producers and scroungers to persist either alone or together are distinguished from those in which they cannot. Producer persistence depends in general on the distribution of resources and producer movement, whereas scrounger persistence depends on its ability to invade when producers are at steady-state. It is found that (i) both species can persist when the habitat has high productivity, (ii) neither species can persist when the habitat has low productivity, and (iii) slower dispersal of both the producer and scrounger is favored when the habitat has intermediate productivity.
Citation: Chris Cosner, Andrew L. Nevai. Spatial population dynamics in a producer-scrounger model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1591-1607. doi: 10.3934/dcdsb.2015.20.1591
##### References:
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Behav., 27 (1979), 487. doi: 10.1016/0003-3472(79)90185-4. Google Scholar [13] M. Broom and G. D. Ruxton, Evolutionarily stable stealing: Game theory applied to kleptoparasitism,, Behav. Ecol., 9 (1998), 397. Google Scholar [14] M. Broom and G. D. Ruxton, Evolutionarily stable kleptoparasitism: Consequences of different prey type,, Behav. Ecol., 14 (2003), 23. doi: 10.1093/beheco/14.1.23. Google Scholar [15] M. Broom, R. M. Luther and G. D. Ruxton, Resistance is useless? - Extensions to the game theory of kleptoparasitism,, Bull. Math. Biol., 66 (2004), 1645. doi: 10.1016/j.bulm.2004.03.009. Google Scholar [16] M. Broom, R. M. Luther, G. D. Ruxton and J. Rychtar, A game-theoretic model of kleptoparasitic behavior in polymorphic populations,, J. Theor. Biol., 255 (2008), 81. doi: 10.1016/j.jtbi.2008.08.001. Google Scholar [17] M. Broom, M. Crowe, M. Fitzgerald and J. Rychtar, The stochastic modelling of kleptoparasitism using a Markov process,, J. Theor. Biol., 264 (2010), 266. doi: 10.1016/j.jtbi.2010.01.012. Google Scholar [18] K. J. Brown and S. S. Lin, On the existence of positive eigenfunctions for an eigenvalue problem with indefinite weight function,, J. Math. Anal. Appl., 75 (1980), 112. doi: 10.1016/0022-247X(80)90309-1. Google Scholar [19] T. Bugnyar and K. Kotrschal, Scrounging tactics in free-ranging ravens, Corvus corax,, Ethology, 108 (2002), 993. doi: 10.1046/j.1439-0310.2002.00832.x. Google Scholar [20] R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations,, Wiley Ser. Math. Comput. Biol., (2003). doi: 10.1002/0470871296. Google Scholar [21] R. S. Cantrell, C. Cosner and Y. Lou, Movement towards better environments and the evolution of rapid diffusion,, Math. Biosci., 204 (2006), 199. doi: 10.1016/j.mbs.2006.09.003. Google Scholar [22] R. S. Cantrell, C. Cosner and Y. Lou, Advection-mediated coexistence of competing species,, Proc. Roy. Soc. Edinb. A, 137 (2007), 497. doi: 10.1017/S0308210506000047. Google Scholar [23] R.S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal in heterogeneous landscapes,, in Spatial Ecology (eds. R. S. Cantrell, (2009), 213. doi: 10.1201/9781420059861.ch11. Google Scholar [24] R. S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal and ideal free distribution,, Math. Biosci. Eng., 7 (2010), 17. doi: 10.3934/mbe.2010.7.17. Google Scholar [25] R. S. Cantrell, C. Cosner and V. Hutson, Permanence in ecological systems with diffusion,, Proc. Royal Soc. Edinburgh, 123 (1993), 533. doi: 10.1017/S0308210500025877. Google Scholar [26] X. Chen, K-Y. Lam and Y. Lou, Dynamics of a reaction-diffusion-advection model for two competing species,, Disc. Cont. Dyn. Syst. A, 32 (2012), 3841. doi: 10.3934/dcds.2012.32.3841. Google Scholar [27] C. Cosner, Reaction-diffusion equations and ecological modeling,, in Tutorials in Mathematical Biosciences. IV, (2008), 77. doi: 10.1007/978-3-540-74331-6_3. Google Scholar [28] C. Cosner and Y. Lou, When does movement toward better environment benefit a population?,, J. Math. Anal. Appl., 277 (2003), 489. doi: 10.1016/S0022-247X(02)00575-9. Google Scholar [29] C. Cosner, A. L. Nevai and Z. Shuai, In, preparation., (). Google Scholar [30] J. Dockery, V. Hutson, K. Mischaikow and M. Pernarowski, The evolution of slow dispersal rates: A reaction-diffusion model,, J. Math. Biol., 37 (1998), 61. doi: 10.1007/s002850050120. Google Scholar [31] L. C. Evans, Partial Differential Equations,, Grad. Stud. Math., (1998). Google Scholar [32] L.-A. Giraldeau and T. Caraco, Social Foraging Theory,, Princeton University Press, (2000). Google Scholar [33] B. S. Goh, Global stability in many-species systems,, Am. Nat., 111 (1977), 135. doi: 10.1086/283144. Google Scholar [34] J. K. Hale and P. Waltman, Persistence in infinite dimensional systems,, SIAM J. Math. Anal., 20 (1989), 388. doi: 10.1137/0520025. Google Scholar [35] I. M. Hamilton, Kleptoparasitism and the distribution of unequal competitors,, Behav. Ecol., 13 (2002), 260. doi: 10.1093/beheco/13.2.260. Google Scholar [36] A. Hastings, Can spatial variation alone lead to selection for dispersal?,, Theor. Pop. Biol., 24 (1983), 244. doi: 10.1016/0040-5809(83)90027-8. Google Scholar [37] D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lecture Notes in Mathematics, (1981). Google Scholar [38] E. Iyengar, Kleptoparasitic interactions throughout the animal kingdom and a re-evaluation, based on participant mobility, of the conditions promoting the evolution of kleptoparasitism,, Biol. J. Linnean Soc., 93 (2008), 745. doi: 10.1111/j.1095-8312.2008.00954.x. Google Scholar [39] R. M. Luther and M. Broom, Rapid convergence to an equilibrium state in kleptoparasitic populations,, J. Math. Biol., 48 (2004), 325. doi: 10.1007/s00285-003-0237-3. Google Scholar [40] R. M. Luther, M. Broom and G. D. Ruxton, Is food worth fighting for? ESS's in mixed populations of kleptoparasites and foragers,, Bull. Math. Biol., 69 (2007), 1121. doi: 10.1007/s11538-005-9052-x. Google Scholar [41] U. Luttge, Vascular Plants as Epiphytes,, Springer-Verlag, (1989). Google Scholar [42] X. Mora, Semilinear parabolic problems define semiflows on $C^k$ spaces,, Trans. Amer. Math. Soc., 278 (1983), 21. doi: 10.2307/1999300. Google Scholar [43] J. Morand-Ferron, G.-M. Wu and L.-A. Giraldeau, Persistent individual differences in tactic use in a producer-scrounger game are group dependent,, Anim. Behav., 82 (2011), 811. Google Scholar [44] J. Morgan, Boundedness and decay results for reaction-diffusion systems,, SIAM J. Math. Anal., 21 (1990), 1172. doi: 10.1137/0521064. Google Scholar [45] K. Mottley and L.-A. Giraldeau, Experimental evidence that group foragers can converge on predicted producer-scrounger equilibria,, Anim. Behav., 60 (2000), 341. doi: 10.1006/anbe.2000.1474. Google Scholar [46] Y. Ohtsuka and Y. Toquenaga, The patch distributed producer-scrounger game,, J. Theor. Biol., 260 (2009), 261. doi: 10.1016/j.jtbi.2009.06.002. Google Scholar [47] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar [48] F. J. Richard, A. Dejean and J. P. Lachaud, Sugary food robbing in ants: A case of temporal cleptobiosis,, C. R. Biologies, 327 (2004), 509. doi: 10.1016/j.crvi.2004.03.002. Google Scholar [49] G. D. Ruxton, Foraging in flocks: Non-spatial models may neglect important costs,, Ecol. Model., 82 (1995), 277. doi: 10.1016/0304-3800(94)00098-3. Google Scholar [50] S. Senn and P. Hess, On positive solutions of a linear elliptic eigenvalue problem with Neumann boundary conditions,, Math. Ann., 258 (1982), 459. doi: 10.1007/BF01453979. Google Scholar [51] N. Tania, B. Vanderlii, J. P. Heath and L. Edelstein-Keshet, Role of social interactions in dynamic patterns of resource patches and forager aggregation,, PNAS, 109 (2012), 11228. doi: 10.1073/pnas.1201739109. Google Scholar [52] J. H. M. Thornley and I. R. Johnson, Plant and Crop Modelling - A Mathematical Approach to Plant and Crop Physiology,, The Blackburn Press, (2000). Google Scholar [53] W. L. Vickery, L.-A. Giraldeau, J. J. Templeton, D. L. Kramer and C. A. Chapman, Producers, scroungers, and group foraging,, Am. Nat., 137 (1991), 847. doi: 10.1086/285197. Google Scholar [54] F. Vollrath, Behaviour of the kleptoparasitic spider Argyrodes elevatus (Araneae, Theridiidae),, Anim. Behav., 27 (1979), 519. doi: 10.1016/0003-3472(79)90186-6. Google Scholar

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##### References:
 [1] H. Amann, Global existence for semilinear parabolic systems,, J. Reine Angew. Math., 360 (1985), 47. doi: 10.1515/crll.1985.360.47. Google Scholar [2] H. Amann, Quasilinear evolution equations and parabolic systems,, Trans. Amer. Math. Soc, 293 (1986), 191. doi: 10.1090/S0002-9947-1986-0814920-4. Google Scholar [3] H. Amann, Dynamic theory of quasilinear parabolic equations I: Abstract evolution equations,, Nonlinear Analysis TMA, 12 (1988), 895. doi: 10.1016/0362-546X(88)90073-9. Google Scholar [4] H. Amann, Dynamic theory of quasilinear parabolic systems III: Global existence,, Math. Z., 202 (1989), 219. doi: 10.1007/BF01215256. Google Scholar [5] H. Amann, Dynamic theory of quasilinear parabolic equations II: Reaction-diffusion systems,, Differential and Integral Equations, 3 (1990), 13. Google Scholar [6] C. J. Barnard, Producers and Scroungers: Strategies of Exploitation and Parasitism,, Chapman and Hall, (1984). Google Scholar [7] C. J. Barnard and R. M. Sibly, Producers and scroungers: A general model and its application to captive flocks of house sparrows,, Anim. Behav., 29 (1981), 543. doi: 10.1016/S0003-3472(81)80117-0. Google Scholar [8] Z. Barta, R. Flynn and L.-A. Giraldeau, Geometry for a selfish foraging group: A genetic algorithm approach,, Proc Roy Soc Lond B., 264 (1997), 1233. doi: 10.1098/rspb.1997.0170. Google Scholar [9] G. Beauchamp, Learning rules for social foragers: Implications for the producer-scrounger game and ideal free distribution theory,, J. Theor. Biol., 207 (2000), 21. doi: 10.1006/jtbi.2000.2153. Google Scholar [10] G. Beauchamp, A spatial model of producing and scrounging,, Anim. Behav., 76 (2008), 1935. doi: 10.1016/j.anbehav.2008.08.017. Google Scholar [11] J. Bilotti and J. P. LaSalle, Periodic dissipative processes,, Bull. Amer. Math. Soc., 77 (1971), 1082. doi: 10.1090/S0002-9904-1971-12879-3. Google Scholar [12] H. J. Brockmann and C. J. Barnard, Kleptoparasitism in birds,, Anim. Behav., 27 (1979), 487. doi: 10.1016/0003-3472(79)90185-4. Google Scholar [13] M. Broom and G. D. Ruxton, Evolutionarily stable stealing: Game theory applied to kleptoparasitism,, Behav. Ecol., 9 (1998), 397. Google Scholar [14] M. Broom and G. D. Ruxton, Evolutionarily stable kleptoparasitism: Consequences of different prey type,, Behav. Ecol., 14 (2003), 23. doi: 10.1093/beheco/14.1.23. Google Scholar [15] M. Broom, R. M. Luther and G. D. Ruxton, Resistance is useless? - Extensions to the game theory of kleptoparasitism,, Bull. Math. Biol., 66 (2004), 1645. doi: 10.1016/j.bulm.2004.03.009. Google Scholar [16] M. Broom, R. M. Luther, G. D. Ruxton and J. Rychtar, A game-theoretic model of kleptoparasitic behavior in polymorphic populations,, J. Theor. Biol., 255 (2008), 81. doi: 10.1016/j.jtbi.2008.08.001. Google Scholar [17] M. Broom, M. Crowe, M. Fitzgerald and J. Rychtar, The stochastic modelling of kleptoparasitism using a Markov process,, J. Theor. Biol., 264 (2010), 266. doi: 10.1016/j.jtbi.2010.01.012. Google Scholar [18] K. J. Brown and S. S. Lin, On the existence of positive eigenfunctions for an eigenvalue problem with indefinite weight function,, J. Math. Anal. Appl., 75 (1980), 112. doi: 10.1016/0022-247X(80)90309-1. Google Scholar [19] T. Bugnyar and K. Kotrschal, Scrounging tactics in free-ranging ravens, Corvus corax,, Ethology, 108 (2002), 993. doi: 10.1046/j.1439-0310.2002.00832.x. Google Scholar [20] R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations,, Wiley Ser. Math. Comput. Biol., (2003). doi: 10.1002/0470871296. Google Scholar [21] R. S. Cantrell, C. Cosner and Y. Lou, Movement towards better environments and the evolution of rapid diffusion,, Math. Biosci., 204 (2006), 199. doi: 10.1016/j.mbs.2006.09.003. Google Scholar [22] R. S. Cantrell, C. Cosner and Y. Lou, Advection-mediated coexistence of competing species,, Proc. Roy. Soc. Edinb. A, 137 (2007), 497. doi: 10.1017/S0308210506000047. Google Scholar [23] R.S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal in heterogeneous landscapes,, in Spatial Ecology (eds. R. S. Cantrell, (2009), 213. doi: 10.1201/9781420059861.ch11. Google Scholar [24] R. S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal and ideal free distribution,, Math. Biosci. Eng., 7 (2010), 17. doi: 10.3934/mbe.2010.7.17. Google Scholar [25] R. S. Cantrell, C. Cosner and V. Hutson, Permanence in ecological systems with diffusion,, Proc. Royal Soc. Edinburgh, 123 (1993), 533. doi: 10.1017/S0308210500025877. Google Scholar [26] X. Chen, K-Y. Lam and Y. Lou, Dynamics of a reaction-diffusion-advection model for two competing species,, Disc. Cont. Dyn. Syst. A, 32 (2012), 3841. doi: 10.3934/dcds.2012.32.3841. Google Scholar [27] C. Cosner, Reaction-diffusion equations and ecological modeling,, in Tutorials in Mathematical Biosciences. IV, (2008), 77. doi: 10.1007/978-3-540-74331-6_3. Google Scholar [28] C. Cosner and Y. Lou, When does movement toward better environment benefit a population?,, J. Math. Anal. Appl., 277 (2003), 489. doi: 10.1016/S0022-247X(02)00575-9. Google Scholar [29] C. Cosner, A. L. Nevai and Z. Shuai, In, preparation., (). Google Scholar [30] J. Dockery, V. Hutson, K. Mischaikow and M. Pernarowski, The evolution of slow dispersal rates: A reaction-diffusion model,, J. Math. Biol., 37 (1998), 61. doi: 10.1007/s002850050120. Google Scholar [31] L. C. Evans, Partial Differential Equations,, Grad. Stud. Math., (1998). Google Scholar [32] L.-A. Giraldeau and T. Caraco, Social Foraging Theory,, Princeton University Press, (2000). Google Scholar [33] B. S. Goh, Global stability in many-species systems,, Am. Nat., 111 (1977), 135. doi: 10.1086/283144. Google Scholar [34] J. K. Hale and P. Waltman, Persistence in infinite dimensional systems,, SIAM J. Math. Anal., 20 (1989), 388. doi: 10.1137/0520025. Google Scholar [35] I. M. Hamilton, Kleptoparasitism and the distribution of unequal competitors,, Behav. Ecol., 13 (2002), 260. doi: 10.1093/beheco/13.2.260. Google Scholar [36] A. Hastings, Can spatial variation alone lead to selection for dispersal?,, Theor. Pop. Biol., 24 (1983), 244. doi: 10.1016/0040-5809(83)90027-8. Google Scholar [37] D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lecture Notes in Mathematics, (1981). Google Scholar [38] E. Iyengar, Kleptoparasitic interactions throughout the animal kingdom and a re-evaluation, based on participant mobility, of the conditions promoting the evolution of kleptoparasitism,, Biol. J. Linnean Soc., 93 (2008), 745. doi: 10.1111/j.1095-8312.2008.00954.x. Google Scholar [39] R. M. Luther and M. Broom, Rapid convergence to an equilibrium state in kleptoparasitic populations,, J. Math. Biol., 48 (2004), 325. doi: 10.1007/s00285-003-0237-3. Google Scholar [40] R. M. Luther, M. Broom and G. D. Ruxton, Is food worth fighting for? ESS's in mixed populations of kleptoparasites and foragers,, Bull. Math. Biol., 69 (2007), 1121. doi: 10.1007/s11538-005-9052-x. Google Scholar [41] U. Luttge, Vascular Plants as Epiphytes,, Springer-Verlag, (1989). Google Scholar [42] X. Mora, Semilinear parabolic problems define semiflows on $C^k$ spaces,, Trans. Amer. Math. Soc., 278 (1983), 21. doi: 10.2307/1999300. Google Scholar [43] J. Morand-Ferron, G.-M. Wu and L.-A. Giraldeau, Persistent individual differences in tactic use in a producer-scrounger game are group dependent,, Anim. Behav., 82 (2011), 811. Google Scholar [44] J. Morgan, Boundedness and decay results for reaction-diffusion systems,, SIAM J. Math. Anal., 21 (1990), 1172. doi: 10.1137/0521064. Google Scholar [45] K. Mottley and L.-A. Giraldeau, Experimental evidence that group foragers can converge on predicted producer-scrounger equilibria,, Anim. Behav., 60 (2000), 341. doi: 10.1006/anbe.2000.1474. Google Scholar [46] Y. Ohtsuka and Y. Toquenaga, The patch distributed producer-scrounger game,, J. Theor. Biol., 260 (2009), 261. doi: 10.1016/j.jtbi.2009.06.002. Google Scholar [47] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar [48] F. J. Richard, A. Dejean and J. P. Lachaud, Sugary food robbing in ants: A case of temporal cleptobiosis,, C. R. Biologies, 327 (2004), 509. doi: 10.1016/j.crvi.2004.03.002. Google Scholar [49] G. D. Ruxton, Foraging in flocks: Non-spatial models may neglect important costs,, Ecol. Model., 82 (1995), 277. doi: 10.1016/0304-3800(94)00098-3. Google Scholar [50] S. Senn and P. Hess, On positive solutions of a linear elliptic eigenvalue problem with Neumann boundary conditions,, Math. Ann., 258 (1982), 459. doi: 10.1007/BF01453979. Google Scholar [51] N. Tania, B. Vanderlii, J. P. Heath and L. Edelstein-Keshet, Role of social interactions in dynamic patterns of resource patches and forager aggregation,, PNAS, 109 (2012), 11228. doi: 10.1073/pnas.1201739109. Google Scholar [52] J. H. M. Thornley and I. R. Johnson, Plant and Crop Modelling - A Mathematical Approach to Plant and Crop Physiology,, The Blackburn Press, (2000). Google Scholar [53] W. L. Vickery, L.-A. Giraldeau, J. J. Templeton, D. L. Kramer and C. A. Chapman, Producers, scroungers, and group foraging,, Am. Nat., 137 (1991), 847. doi: 10.1086/285197. Google Scholar [54] F. Vollrath, Behaviour of the kleptoparasitic spider Argyrodes elevatus (Araneae, Theridiidae),, Anim. Behav., 27 (1979), 519. doi: 10.1016/0003-3472(79)90186-6. Google Scholar
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