# 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 and Continuous Dynamical Systems - B, 2015, 20 (6) : 1591-1607. doi: 10.3934/dcdsb.2015.20.1591
##### References:
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Sibly, Producers and scroungers: A general model and its application to captive flocks of house sparrows, Anim. Behav., 29 (1981), 543-550. doi: 10.1016/S0003-3472(81)80117-0. [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), 1233-1238. doi: 10.1098/rspb.1997.0170. [9] G. Beauchamp, Learning rules for social foragers: Implications for the producer-scrounger game and ideal free distribution theory, J. Theor. Biol., 207 (2000), 21-35. doi: 10.1006/jtbi.2000.2153. [10] G. Beauchamp, A spatial model of producing and scrounging, Anim. Behav., 76 (2008), 1935-1942. doi: 10.1016/j.anbehav.2008.08.017. [11] J. Bilotti and J. P. LaSalle, Periodic dissipative processes, Bull. Amer. Math. Soc., 77 (1971), 1082-1088. doi: 10.1090/S0002-9904-1971-12879-3. [12] H. J. Brockmann and C. J. Barnard, Kleptoparasitism in birds, Anim. Behav., 27 (1979), 487-514. doi: 10.1016/0003-3472(79)90185-4. [13] M. Broom and G. D. Ruxton, Evolutionarily stable stealing: Game theory applied to kleptoparasitism, Behav. Ecol., 9 (1998), 397-403. [14] M. Broom and G. D. Ruxton, Evolutionarily stable kleptoparasitism: Consequences of different prey type, Behav. Ecol., 14 (2003), 23-33. doi: 10.1093/beheco/14.1.23. [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), 1645-1658. doi: 10.1016/j.bulm.2004.03.009. [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), 81-91. doi: 10.1016/j.jtbi.2008.08.001. [17] M. Broom, M. Crowe, M. Fitzgerald and J. Rychtar, The stochastic modelling of kleptoparasitism using a Markov process, J. Theor. Biol., 264 (2010), 266-272. doi: 10.1016/j.jtbi.2010.01.012. [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), 112-120. doi: 10.1016/0022-247X(80)90309-1. [19] T. Bugnyar and K. Kotrschal, Scrounging tactics in free-ranging ravens, Corvus corax, Ethology, 108 (2002), 993-1009. doi: 10.1046/j.1439-0310.2002.00832.x. [20] R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Ser. Math. Comput. Biol., John Wiley & Sons, Ltd., Chichester, 2003. doi: 10.1002/0470871296. [21] R. S. Cantrell, C. Cosner and Y. Lou, Movement towards better environments and the evolution of rapid diffusion, Math. Biosci., 204 (2006), 199-214. doi: 10.1016/j.mbs.2006.09.003. [22] R. S. Cantrell, C. Cosner and Y. Lou, Advection-mediated coexistence of competing species, Proc. Roy. Soc. Edinb. A, 137 (2007), 497-518. doi: 10.1017/S0308210506000047. [23] R.S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal in heterogeneous landscapes, in Spatial Ecology (eds. R. S. Cantrell, C. Cosner and S. Ruan), Mathematical and Computational Biology Series, Chapman Hall/CRC Press, 2009, 213-229. doi: 10.1201/9781420059861.ch11. [24] R. S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal and ideal free distribution, Math. Biosci. Eng., 7 (2010), 17-36. doi: 10.3934/mbe.2010.7.17. [25] R. S. Cantrell, C. Cosner and V. Hutson, Permanence in ecological systems with diffusion, Proc. Royal Soc. Edinburgh, 123 (1993), 533-559. doi: 10.1017/S0308210500025877. [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), 3841-3859. doi: 10.3934/dcds.2012.32.3841. [27] C. Cosner, Reaction-diffusion equations and ecological modeling, in Tutorials in Mathematical Biosciences. IV, Lecture Notes in Mathematics, {1922}, Springer, Berlin, 2008, 77-115. doi: 10.1007/978-3-540-74331-6_3. [28] C. Cosner and Y. Lou, When does movement toward better environment benefit a population?, J. Math. Anal. Appl., 277 (2003), 489-503. doi: 10.1016/S0022-247X(02)00575-9. [29] C. Cosner, A. L. Nevai and Z. Shuai, In, preparation., (). [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), 61-83. doi: 10.1007/s002850050120. [31] L. C. Evans, Partial Differential Equations, Grad. Stud. Math., 19, Amer. Math. Soc., Providence, RI, 1998. [32] L.-A. Giraldeau and T. Caraco, Social Foraging Theory, Princeton University Press, 2000. [33] B. S. Goh, Global stability in many-species systems, Am. Nat., 111 (1977), 135-143. doi: 10.1086/283144. [34] J. K. Hale and P. Waltman, Persistence in infinite dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395. doi: 10.1137/0520025. [35] I. M. Hamilton, Kleptoparasitism and the distribution of unequal competitors, Behav. Ecol., 13 (2002), 260-267. doi: 10.1093/beheco/13.2.260. [36] A. Hastings, Can spatial variation alone lead to selection for dispersal?, Theor. Pop. Biol., 24 (1983), 244-251. doi: 10.1016/0040-5809(83)90027-8. [37] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. [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), 745-762. doi: 10.1111/j.1095-8312.2008.00954.x. [39] R. M. Luther and M. Broom, Rapid convergence to an equilibrium state in kleptoparasitic populations, J. Math. Biol., 48 (2004), 325-339. doi: 10.1007/s00285-003-0237-3. [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), 1121-1146. doi: 10.1007/s11538-005-9052-x. [41] U. Luttge, Vascular Plants as Epiphytes, Springer-Verlag, New York, 1989. [42] X. Mora, Semilinear parabolic problems define semiflows on $C^k$ spaces, Trans. Amer. Math. Soc., 278 (1983), 21-55. doi: 10.2307/1999300. [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), 811-816. [44] J. Morgan, Boundedness and decay results for reaction-diffusion systems, SIAM J. Math. Anal., 21 (1990), 1172-1189. doi: 10.1137/0521064. [45] K. Mottley and L.-A. Giraldeau, Experimental evidence that group foragers can converge on predicted producer-scrounger equilibria, Anim. Behav., 60 (2000), 341-350. doi: 10.1006/anbe.2000.1474. [46] Y. Ohtsuka and Y. Toquenaga, The patch distributed producer-scrounger game, J. Theor. Biol., 260 (2009), 261-266. doi: 10.1016/j.jtbi.2009.06.002. [47] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, Berlin, 1983. doi: 10.1007/978-1-4612-5561-1. [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), 509-517. doi: 10.1016/j.crvi.2004.03.002. [49] G. D. Ruxton, Foraging in flocks: Non-spatial models may neglect important costs, Ecol. Model., 82 (1995), 277-285. doi: 10.1016/0304-3800(94)00098-3. [50] S. Senn and P. Hess, On positive solutions of a linear elliptic eigenvalue problem with Neumann boundary conditions, Math. Ann., 258 (1982), 459-470. doi: 10.1007/BF01453979. [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), 11228-11233. doi: 10.1073/pnas.1201739109. [52] J. H. M. Thornley and I. R. Johnson, Plant and Crop Modelling - A Mathematical Approach to Plant and Crop Physiology, The Blackburn Press, New Jersey, 2000. [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), 847-863. doi: 10.1086/285197. [54] F. Vollrath, Behaviour of the kleptoparasitic spider Argyrodes elevatus (Araneae, Theridiidae), Anim. Behav., 27 (1979), 519-521. doi: 10.1016/0003-3472(79)90186-6.

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##### References:
 [1] H. Amann, Global existence for semilinear parabolic systems, J. Reine Angew. Math., 360 (1985), 47-83. doi: 10.1515/crll.1985.360.47. [2] H. Amann, Quasilinear evolution equations and parabolic systems, Trans. Amer. Math. Soc, 293 (1986), 191-227. doi: 10.1090/S0002-9947-1986-0814920-4. [3] H. Amann, Dynamic theory of quasilinear parabolic equations I: Abstract evolution equations, Nonlinear Analysis TMA, 12 (1988), 895-919. doi: 10.1016/0362-546X(88)90073-9. [4] H. Amann, Dynamic theory of quasilinear parabolic systems III: Global existence, Math. Z., 202 (1989), 219-250. doi: 10.1007/BF01215256. [5] H. Amann, Dynamic theory of quasilinear parabolic equations II: Reaction-diffusion systems, Differential and Integral Equations, 3 (1990), 13-75. [6] C. J. Barnard, Producers and Scroungers: Strategies of Exploitation and Parasitism, Chapman and Hall, 1984. [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), 543-550. doi: 10.1016/S0003-3472(81)80117-0. [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), 1233-1238. doi: 10.1098/rspb.1997.0170. [9] G. Beauchamp, Learning rules for social foragers: Implications for the producer-scrounger game and ideal free distribution theory, J. Theor. Biol., 207 (2000), 21-35. doi: 10.1006/jtbi.2000.2153. [10] G. Beauchamp, A spatial model of producing and scrounging, Anim. Behav., 76 (2008), 1935-1942. doi: 10.1016/j.anbehav.2008.08.017. [11] J. Bilotti and J. P. LaSalle, Periodic dissipative processes, Bull. Amer. Math. Soc., 77 (1971), 1082-1088. doi: 10.1090/S0002-9904-1971-12879-3. [12] H. J. Brockmann and C. J. Barnard, Kleptoparasitism in birds, Anim. Behav., 27 (1979), 487-514. doi: 10.1016/0003-3472(79)90185-4. [13] M. Broom and G. D. Ruxton, Evolutionarily stable stealing: Game theory applied to kleptoparasitism, Behav. Ecol., 9 (1998), 397-403. [14] M. Broom and G. D. Ruxton, Evolutionarily stable kleptoparasitism: Consequences of different prey type, Behav. Ecol., 14 (2003), 23-33. doi: 10.1093/beheco/14.1.23. [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), 1645-1658. doi: 10.1016/j.bulm.2004.03.009. [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), 81-91. doi: 10.1016/j.jtbi.2008.08.001. [17] M. Broom, M. Crowe, M. Fitzgerald and J. Rychtar, The stochastic modelling of kleptoparasitism using a Markov process, J. Theor. Biol., 264 (2010), 266-272. doi: 10.1016/j.jtbi.2010.01.012. [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), 112-120. doi: 10.1016/0022-247X(80)90309-1. [19] T. Bugnyar and K. Kotrschal, Scrounging tactics in free-ranging ravens, Corvus corax, Ethology, 108 (2002), 993-1009. doi: 10.1046/j.1439-0310.2002.00832.x. [20] R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Ser. Math. Comput. Biol., John Wiley & Sons, Ltd., Chichester, 2003. doi: 10.1002/0470871296. [21] R. S. Cantrell, C. Cosner and Y. Lou, Movement towards better environments and the evolution of rapid diffusion, Math. Biosci., 204 (2006), 199-214. doi: 10.1016/j.mbs.2006.09.003. [22] R. S. Cantrell, C. Cosner and Y. Lou, Advection-mediated coexistence of competing species, Proc. Roy. Soc. Edinb. A, 137 (2007), 497-518. doi: 10.1017/S0308210506000047. [23] R.S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal in heterogeneous landscapes, in Spatial Ecology (eds. R. S. Cantrell, C. Cosner and S. Ruan), Mathematical and Computational Biology Series, Chapman Hall/CRC Press, 2009, 213-229. doi: 10.1201/9781420059861.ch11. [24] R. S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal and ideal free distribution, Math. Biosci. Eng., 7 (2010), 17-36. doi: 10.3934/mbe.2010.7.17. [25] R. S. Cantrell, C. Cosner and V. Hutson, Permanence in ecological systems with diffusion, Proc. Royal Soc. Edinburgh, 123 (1993), 533-559. doi: 10.1017/S0308210500025877. [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), 3841-3859. doi: 10.3934/dcds.2012.32.3841. [27] C. Cosner, Reaction-diffusion equations and ecological modeling, in Tutorials in Mathematical Biosciences. IV, Lecture Notes in Mathematics, {1922}, Springer, Berlin, 2008, 77-115. doi: 10.1007/978-3-540-74331-6_3. [28] C. Cosner and Y. Lou, When does movement toward better environment benefit a population?, J. Math. Anal. Appl., 277 (2003), 489-503. doi: 10.1016/S0022-247X(02)00575-9. [29] C. Cosner, A. L. Nevai and Z. Shuai, In, preparation., (). [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), 61-83. doi: 10.1007/s002850050120. [31] L. C. Evans, Partial Differential Equations, Grad. Stud. Math., 19, Amer. Math. Soc., Providence, RI, 1998. [32] L.-A. Giraldeau and T. Caraco, Social Foraging Theory, Princeton University Press, 2000. [33] B. S. Goh, Global stability in many-species systems, Am. Nat., 111 (1977), 135-143. doi: 10.1086/283144. [34] J. K. Hale and P. Waltman, Persistence in infinite dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395. doi: 10.1137/0520025. [35] I. M. Hamilton, Kleptoparasitism and the distribution of unequal competitors, Behav. Ecol., 13 (2002), 260-267. doi: 10.1093/beheco/13.2.260. [36] A. Hastings, Can spatial variation alone lead to selection for dispersal?, Theor. Pop. Biol., 24 (1983), 244-251. doi: 10.1016/0040-5809(83)90027-8. [37] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. [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), 745-762. doi: 10.1111/j.1095-8312.2008.00954.x. [39] R. M. Luther and M. Broom, Rapid convergence to an equilibrium state in kleptoparasitic populations, J. Math. Biol., 48 (2004), 325-339. doi: 10.1007/s00285-003-0237-3. [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), 1121-1146. doi: 10.1007/s11538-005-9052-x. [41] U. Luttge, Vascular Plants as Epiphytes, Springer-Verlag, New York, 1989. [42] X. Mora, Semilinear parabolic problems define semiflows on $C^k$ spaces, Trans. Amer. Math. Soc., 278 (1983), 21-55. doi: 10.2307/1999300. [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), 811-816. [44] J. Morgan, Boundedness and decay results for reaction-diffusion systems, SIAM J. Math. Anal., 21 (1990), 1172-1189. doi: 10.1137/0521064. [45] K. Mottley and L.-A. Giraldeau, Experimental evidence that group foragers can converge on predicted producer-scrounger equilibria, Anim. Behav., 60 (2000), 341-350. doi: 10.1006/anbe.2000.1474. [46] Y. Ohtsuka and Y. Toquenaga, The patch distributed producer-scrounger game, J. Theor. Biol., 260 (2009), 261-266. doi: 10.1016/j.jtbi.2009.06.002. [47] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, Berlin, 1983. doi: 10.1007/978-1-4612-5561-1. [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), 509-517. doi: 10.1016/j.crvi.2004.03.002. [49] G. D. Ruxton, Foraging in flocks: Non-spatial models may neglect important costs, Ecol. Model., 82 (1995), 277-285. doi: 10.1016/0304-3800(94)00098-3. [50] S. Senn and P. Hess, On positive solutions of a linear elliptic eigenvalue problem with Neumann boundary conditions, Math. Ann., 258 (1982), 459-470. doi: 10.1007/BF01453979. [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), 11228-11233. doi: 10.1073/pnas.1201739109. [52] J. H. M. Thornley and I. R. Johnson, Plant and Crop Modelling - A Mathematical Approach to Plant and Crop Physiology, The Blackburn Press, New Jersey, 2000. [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), 847-863. doi: 10.1086/285197. [54] F. Vollrath, Behaviour of the kleptoparasitic spider Argyrodes elevatus (Araneae, Theridiidae), Anim. Behav., 27 (1979), 519-521. doi: 10.1016/0003-3472(79)90186-6.
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