Hybrid approach to modeling spatial dynamics of systems with generalist predators. (English) Zbl 1406.92661
Summary: We consider hybrid spatial modeling approaches for ecological systems with a generalist predator utilizing a prey and either a second prey or an allochthonous resource. While spatial dispersion of populations is often modeled via stepping-stone (discrete spatial patches) or continuum (one connected spatial domain) formulations, we shall be interested in hybrid approaches which we use to reduce the dimension of certain components of the spatial domain, obtaining either a continuum model of varying spatial dimensions, or a mixed stepping-stone-continuum model. This approach results in models consisting of partial differential equations for some of the species which are coupled via reactive boundary conditions to lower dimensional partial differential equations or ordinary differential equations for the other species. In order to demonstrate the use of this approach, we consider two case studies. In the first case study, we consider a one-predator two-prey interaction between beavers, wolves and white-tailed deer in Voyageurs National Park. In the second case study, we consider predator-prey-allochthonous resource interactions between bears, berries and salmon on Kodiak Island. For each case study, we compare the results from the hybrid modeling approach with corresponding stepping-stone and continuum model results, highlighting benefits and limitations of the method. In some cases, we find that the hybrid modeling approach allows for solutions which are easier to simulate (akin to stepping-stone models) while maintaining seemingly more realistic spatial dynamics (akin to full continuum models).
References:
| [1] | Abrams, P. A., The effects of switching behavior on the evolutionary diversification of generalist consumers, Am. Nat., 168, 5, 645-659 (2006) |
| [2] | Alonso, D.; Bartumeus, F.; Catalan, J., Mutual interference between predators can give rise to turing spatial patterns, Ecology, 83, 1, 28-34 (2002) |
| [3] | Amarasekare, P., Spatial dynamics of communities with intraguild predation: the role of dispersal strategies, Am. Nat., 170, 6, 819-831 (2007) |
| [4] | Andersen, M., Properties of some density-dependent integrodifference equation population models, Math. Biosci., 104, 1, 135-157 (1991) · Zbl 0726.92026 |
| [5] | Anderson, R., A hybrid mathematical model of solid tumour invasion: the importance of cell adhesion, Math. Med. Biol., 22, 2, 163-186 (2005) · Zbl 1073.92013 |
| [6] | Armsworth, P. R.; Roughgarden, J. E., The impact of directed versus random movement on population dynamics and biodiversity patterns, Am. Nat., 165, 4, 449-465 (2005) |
| [7] | Bacaër, N., Verhulst and the Logistic Equation (1838), A Short History of Mathematical Population Dynamics, 35-39 (2011), Springer |
| [8] | Baggio, J. A.; Salau, K.; Janssen, M. A.; Schoon, M. L.; Bodin, Ö., Landscape connectivity and predator-prey population dynamics, Landsc. Ecol., 26, 1, 33-45 (2011) |
| [9] | Barnes Jr, V. G., The influence of salmon availability on movements and range of brown bears on southwest kodiak island, Bears, 305-313 (1990) |
| [10] | Bassett, A.; Krause, A. L.; Van Gorder, R. A., Continuous dispersal in a model of predator-prey-subsidy population dynamics, Ecol. Modell., 354, 115-122 (2017) |
| [11] | Baurmann, M.; Gross, T.; Feudel, U., Instabilities in spatially extended predator-prey systems: spatio-temporal patterns in the neighborhood of turing-hopf bifurcations, J. Theor. Biol., 245, 2, 220-229 (2007) · Zbl 1451.92248 |
| [12] | Berns, V. D.; Atwell, G. C.; Boone, D. L., Brown bear movements and habitat use at karluk lake, kodiak island, Bears, 293-296 (1980) |
| [13] | Berns, V. D.; Hensel, R. J., Radio tracking brown bears on kodiak island, Bears, 19-25 (1972) |
| [14] | ter Braak, C. J.; Hanski, I.; Verboom, J., The incidence function approach to modeling of metapopulation dynamics, Model. Spatiotemporal Dyn. Ecol., 167-188 (1998) |
| [15] | Briggs, C. J.; Hoopes, M. F., Stabilizing effects in spatial parasitoid-host and predator-prey models: a review, Theor. Popul. Biol., 65, 3, 299-315 (2004) · Zbl 1109.92047 |
| [16] | Burkey, T. V., Extinction in nature reserves: the effect of fragmentation and the importance of migration between reserve fragments, Oikos, 75-81 (1989) |
| [17] | Cantrell, R. S.; Cosner, C., Spatial ecology via reaction-diffusion equations (2004), John Wiley & Sons: John Wiley & Sons Chichester, UK · Zbl 1058.92041 |
| [18] | Chen, X.; Hambrock, R.; Lou, Y., Evolution of conditional dispersal: a reaction-diffusion-advection model, J. Math. Biol., 57, 3, 361-386 (2008) · Zbl 1141.92040 |
| [19] | Council, N. R., Rock fractures and fluid flow: Contemporary understanding and applications (1996), National Academies Press: National Academies Press Washington, D.C. |
| [20] | Crowder, L.; Norse, E., Essential ecological insights for marine ecosystem-based management and marine spatial planning, Mar. Policy, 32, 5, 772-778 (2008) |
| [21] | Crowley, P. H., Dispersal and the stability of predator-prey interactions, Am. Nat., 118, 5, 673-701 (1981) |
| [22] | Deka, B.; Patra, A.; Tushar, J.; Dubey, B., Stability and hopf-bifurcation in a general gauss type two-prey and one-predator system, Appl. Math. Model., 40, 11-12, 5793-5818 (2016) · Zbl 1465.92087 |
| [23] | El Abdllaoui, A.; Auger, P.; Kooi, B. W.; De la Parra, R. B.; Mchich, R., Effects of density-dependent migrations on stability of a two-patch predator-prey model, Math. Biosci., 210, 1, 335-354 (2007) · Zbl 1129.92064 |
| [24] | Gable, T. D.; Windels, S. K.; Bruggink, J. G.; Homkes, A. T., Where and how wolves (canis lupus) kill beavers (castor canadensis), PLoS ONE, 11, 12, e0165537 (2016) |
| [25] | Gable, T. D.; Windels, S. K.; Romanski, M. C.; Rosell, F., The forgotten prey of an iconic predator: a review of interactions between grey wolves canis lupus and beavers castor spp, Mamm. Rev., 48, 2, 123-138 (2018) |
| [26] | Gamito, S., Growth models and their use in ecological modelling: an application to a fish population, Ecol. Modell., 113, 1-3, 83-94 (1998) |
| [27] | Gantmakher, F. R., The theory of matrices, 131 (1998), American Mathematical Soc · Zbl 0050.24804 |
| [28] | Genkai-Kato, M.; Yamamura, N., Unpalatable prey resolves the paradox of enrichment, Proc. R Society of London B: Biological Sciences, 266, 1425, 1215-1219 (1999) |
| [29] | Gogan, P.; Route, W.; Olexa, E.; Thomas, N.; Kuehn, D.; Podruzny, K., Gray wolves in and adjacent to Voyageurs National Park, Minnesota: Research and synthesis 1987-1991, Technical Report (2004), National Park Service |
| [30] | Goldwyn, E. E.; Hastings, A., When can dispersal synchronize populations?, Theor. Popul. Biol., 73, 3, 395-402 (2008) · Zbl 1210.92051 |
| [31] | Goldwyn, E. E.; Hastings, A., Small heterogeneity has large effects on synchronization of ecological oscillators, Bull. Math. Biol., 71, 1, 130-144 (2009) · Zbl 1169.92044 |
| [32] | Grindrod, P., Models of individual aggregation or clustering in single and multi-species communities, J. Math. Biol., 26, 6, 651-660 (1988) · Zbl 0714.92024 |
| [33] | Guo, Q.; Taper, M.; Schoenberger, M.; Brandle, J., Spatial-temporal population dynamics across species range: from centre to margin, Oikos, 108, 1, 47-57 (2005) |
| [34] | Gurney, W. S.; Veitch, A. R., Self-organization, scale and stability in a spatial predator-prey interaction, Bull. Math. Biol., 62, 1, 61-86 (2000) · Zbl 1323.92168 |
| [35] | Hamilton, C. D.; Kovacs, K. M.; Ims, R. A.; Aars, J.; Lydersen, C., An arctic predator-prey system in flux: climate change impacts on coastal space use by polar bears and ringed seals, J. Anim. Ecol (2017) |
| [36] | Hanski, I., Metapopulation dynamics, Nature, 396, 6706, 41 (1998) |
| [37] | Hassell, M. P.; Comins, H. N.; Mayt, R. M., Spatial structure and chaos in insect population dynamics, Nature, 353, 6341, 255 (1991) |
| [38] | Hauzy, C.; Gauduchon, M.; Hulot, F. D.; Loreau, M., Density-dependent dispersal and relative dispersal affect the stability of predator-prey metacommunities, J. Theor. Biol., 266, 3, 458-469 (2010) · Zbl 1407.92142 |
| [39] | Hauzy, C.; Hulot, F. D.; Gins, A.; Loreau, M., Intra-and interspecific density-dependent dispersal in an aquatic prey-predator system, J. Anim. Ecol., 76, 3, 552-558 (2007) |
| [40] | Holling, C. S., The components of predation as revealed by a study of small-mammal predation of the european pine sawfly1, Can. Entomol., 91, 5, 293-320 (1959) |
| [41] | Holmes, E. E.; Lewis, M. A.; Banks, J.; Veit, R., Partial differential equations in ecology: spatial interactions and population dynamics, Ecology, 75, 1, 17-29 (1994) |
| [42] | Holt, R. D., Spatial heterogeneity, indirect interactions, and the coexistence of prey species, Am. Nat., 124, 3, 377-406 (1984) |
| [43] | Hopper, K. R.; Roush, R. T., Mate finding, dispersal, number released, and the success of biological control introductions, Ecol. Entomol., 18, 4, 321-331 (1993) |
| [44] | Huang, Y.; Diekmann, O., Predator migration in response to prey density: what are the consequences?, J. Math. Biol., 43, 6, 561-581 (2001) · Zbl 0995.92044 |
| [45] | Hutson, V.; Vickers, G., A criterion for permanent coexistence of species, with an application to a two-prey one-predator system, Math Biosci, 63, 2, 253-269 (1983) · Zbl 0524.92023 |
| [46] | Jansen, J. E.; Van Gorder, R. A., Dynamics from a predator-prey-quarry-resource-scavenger model, Theor. Ecol., 11, 19-38 (2018) |
| [47] | Jansen, V. A., Regulation of predator-prey systems through spatial interactions: a possible solution to the paradox of enrichment, Oikos, 384-390 (1995) |
| [48] | Jansen, V. A., The dynamics of two diffusively coupled predator-prey populations, Theor. Popul. Biol., 59, 2, 119-131 (2001) · Zbl 1036.92034 |
| [49] | Jansen, V. A.; Lloyd, A. L., Local stability analysis of spatially homogeneous solutions of multi-patch systems, J. Math. Biol., 41, 3, 232-252 (2000) · Zbl 0982.92032 |
| [50] | Jansen, V. A.; de Roos, A. M., The role of space in reducing predator-prey cycles, Geometr. Ecolog. Interact., 183-201 (2000) |
| [51] | Johnston, C. A., Beavers: Boreal ecosystem engineers (2017), Springer: Springer Cham, Switzerland |
| [52] | Kang, Y., Sasmal, S.K., Messan, K., 2015. A two-patch prey-predator model with dispersal in predators driven by the strength of predation. arXiv:1505.03820; Kang, Y., Sasmal, S.K., Messan, K., 2015. A two-patch prey-predator model with dispersal in predators driven by the strength of predation. arXiv:1505.03820 · Zbl 1417.37282 |
| [53] | Kareiva, P., Population dynamics in spatially complex environments: theory and data, Philos. Trans., 330, 1257, 175-190 (1990) |
| [54] | Kurowski, L.; Krause, A. L.; Mizuguchi, H.; Grindrod, P.; Van Gorder, R. A., Two-species migration and clustering in two-dimensional domains, Bull. Math. Biol., 79, 10, 2302-2333 (2017) · Zbl 1378.92058 |
| [55] | Levin, S., Complex adaptive systems: exploring the known, the unknown and the unknowable, Bull. Amer. Math. Soc., 40, 1, 3-19 (2003) · Zbl 1015.92001 |
| [56] | Levin, S. A., Dispersion and population interactions, Am. Nat., 108, 960, 207-228 (1974) |
| [57] | Levin, S. A., Population dynamic models in heterogeneous environments, Annu. Rev. Ecol. Syst., 7, 1, 287-310 (1976) |
| [58] | Levy, D.; Harrington, H. A.; Van Gorder, R. A., Role of seasonality on predator-prey-subsidy population dynamics, J. Theor. Biol., 396, 163-181 (2016) · Zbl 1343.92422 |
| [59] | Lewis, M. A.; Maini, P. K.; Petrovskii, S. V., Dispersal, individual movement and spatial ecology, Lecture Notes Math. (Mathematics Bioscience Series), 2071 (2013) · Zbl 1264.92051 |
| [60] | Liebhold, A.; Koenig, W. D.; Bjørnstad, O. N., Spatial synchrony in population dynamics, Annu. Rev. Ecol. Evol. Syst., 35, 467-490 (2004) |
| [61] | Lima, S. L., Putting predators back into behavioral predator-prey interactions, Trend. Ecol. Evolut., 17, 2, 70-75 (2002) |
| [62] | Maciel, G. A.; Kraenkel, R. A., How population loss through habitat boundaries determines the dynamics of a predator-prey system, Ecol. Complexity, 20, 33-42 (2014) |
| [63] | Malthus, T. R., An essay on the principle of population: or, A view of its past and present effects on human happiness, Ninth Edition (1988), Reeves & Turner: Reeves & Turner London, UK |
| [64] | Metz, H., The geometry of ecological interactions: Simplifying spatial complexity (2000), Cambridge University Press: Cambridge University Press Cambridge, UK |
| [65] | Moinfar, A.; Varavei, A.; Sepehrnoori, K.; Johns, R. T., Development of a coupled dual continuum and discrete fracture model for the simulation of unconventional reservoirs, SPE Reservoir Simulation Symposium (2013), Society of Petroleum Engineers |
| [66] | Mougi, A., Coevolution in a one predator-two prey system, PLoS ONE, 5, 11, e13887 (2010) |
| [67] | Murray, J. D., Mathematical Biology I, An Introduction (2002), Springer-Verlag: Springer-Verlag New York · Zbl 1006.92001 |
| [68] | Nadell, C. D.; Drescher, K.; Foster, K. R., Spatial structure, cooperation and competition in biofilms, Nat. Rev. Microbiol., 14, 9, 589 (2016) |
| [69] | Nathan, R., The challenges of studying dispersal, Trend. Ecol. Evolut., 16, 9, 481-483 (2001) |
| [70] | Neubert, M. G.; Klepac, P.; Van den Driessche, P., Stabilizing dispersal delays in predator-prey metapopulation models, Theor. Popul. Biol., 61, 3, 339-347 (2002) · Zbl 1038.92032 |
| [71] | Nevai, A. L.; Van Gorder, R. A., Effect of resource subsidies on predator-prey population dynamics: a mathematical model, J. Biol. Dyn., 6, 2, 891-922 (2012) · Zbl 1447.92356 |
| [72] | Okubo, A.; Levin, S. A., Diffusion and ecological problems: Modern perspectives, 14 (2013), Springer Science & Business Media · Zbl 1027.92022 |
| [73] | Okubo, A.; Maini, P.; Williamson, M.; Murray, J., On the spatial spread of the grey squirrel in britain, Proc. R. Soc. Lond. B, 238, 1291, 113-125 (1989) |
| [74] | Pascual, M., Diffusion-induced chaos in a spatial predator-prey system, Proc. R. Soc. Lond. B, 251, 1330, 1-7 (1993) |
| [75] | Petrovskii, S.; Li, B.-L.; Malchow, H., Transition to spatiotemporal chaos can resolve the paradox of enrichment, Ecol. Complexity, 1, 1, 37-47 (2004) |
| [76] | Pilyugin, S. S.; Medlock, J.; De Leenheer, P., The effectiveness of marine protected areas for predator and prey with varying mobility, Theor. Popul. Biol., 110, 63-77 (2016) · Zbl 1365.92106 |
| [77] | Plitzko, S. J.; Drossel, B., The effect of dispersal between patches on the stability of large trophic food webs, Theor. Ecol., 8, 2, 233-244 (2015) |
| [78] | Ranta, E.; Kaitala, V.; Lundberg, P., The spatial dimension in population fluctuations, Science, 278, 5343, 1621-1623 (1997) |
| [79] | Reimchen, T., Some ecological and evolutionary aspects of bear-salmon interactions in coastal british columbia, Can. J. Zool., 78, 3, 448-457 (2000) |
| [80] | Ripa, J., Analysing the moran effect and dispersal: their significance and interaction in synchronous population dynamics, Oikos, 89, 1, 175-187 (2000) |
| [81] | Roemer, G. W.; Donlan, C. J.; Courchamp, F., Golden eagles, feral pigs, and insular carnivores: how exotic species turn native predators into prey, Proc. Natl. Acad. Sci., 99, 2, 791-796 (2002) |
| [82] | Rosenzweig, M. L., Paradox of enrichment: destabilization of exploitation ecosystems in ecological time, Science, 171, 3969, 385-387 (1971) |
| [83] | Rosenzweig, M. L.; MacArthur, R. H., Graphical representation and stability conditions of predator-prey interactions, Am. Nat., 97, 895, 209-223 (1963) |
| [84] | Roy, S.; Chattopadhyay, J., The stability of ecosystems: a brief overview of the paradox of enrichment, J. Biosci., 32, 2, 421-428 (2007) |
| [85] | Ruokolainen, L.; Abrams, P. A.; McCann, K. S.; Shuter, B. J., The roles of spatial heterogeneity and adaptive movement in stabilizing (or destabilizing) simple metacommunities, J. Theor. Biol., 291, 76-87 (2011) · Zbl 1397.92602 |
| [86] | Scheffer, M.; De Boer, R. J., Implications of spatial heterogeneity for the paradox of enrichment, Ecology, 76, 7, 2270-2277 (1995) |
| [87] | Shen, L.; Van Gorder, R. A., Predator-prey-subsidy population dynamics on stepping-stone domains, J. Theor. Biol., 420, 241-258 (2017) · Zbl 1370.92148 |
| [88] | Shiga, T., Stepping Stone Models in Population Genetics and Population Dynamics, Stochastic Processes in Physics and Engineering, 345-355 (1988), Springer · Zbl 0656.92006 |
| [89] | Suzuki, K.; Yoshida, T., Non-random spatial coupling induces desynchronization, chaos and multistability in a predator-prey-resource system, J. Theor. Biol., 300, 81-90 (2012) · Zbl 1397.92604 |
| [90] | Tilman, D.; Kareiva, P. M., Spatial ecology: The role of space in population dynamics and interspecific interactions, 30 (1997), Princeton University Press |
| [91] | Toyokawa, W., Scrounging by foragers can resolve the paradox of enrichment, R. Soc. Open Sci., 4, 3, 160830 (2017) |
| [92] | Van Kirk, R. W.; Lewis, M. A., Integrodifference models for persistence in fragmented habitats, Bull. Math. Biol., 59, 1, 107 (1997) · Zbl 0923.92031 |
| [93] | Volterra, V., Variations and fluctuations of the number of individuals in animal species living together, ICES J. Mar. Sci., 3, 1, 3-51 (1928) |
| [94] | Weisser, W. W.; Hassell, M. P., Animals’ On the move’stabilize host-parasitoid systems, Proceedings of the Royal Society of London B: Biological Sciences, 263, 1371, 749-754 (1996) |
| [95] | Weisser, W. W.; Jansen, V. A.; Hassell, M. P., The effects of a pool of dispersers on host-parasitoid systems, J. Theor. Biol., 189, 4, 413-425 (1997) |
| [96] | Williamson, M., Biological invasions, 15 (1996), Springer Science & Business Media |
| [97] | Willson, M. F., Mammals as seed-dispersal mutualists in north america, Oikos, 159-176 (1993) |
| [98] | Windels, S. K., Beavers as Engineers of Wildlife Habitat, Beavers: Boreal Ecosystem Engineers, 239-268 (2017), Springer |
| [99] | Yamauchi, A.; Yamamura, N., Effects of defense evolution and diet choice on population dynamics in a one-predator-two-prey system, Ecology, 86, 9, 2513-2524 (2005) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
