Maximum clique analysis (MCA) can approximate landscape carrying capacity (Nk) for populations of territorial wildlife. However, MCA has not been widely adopted for wildlife applications, mainly due to computational constraints and software wildlife biologists may find difficult to use. Moreover, MCA does not incorporate uncertainty into estimates of Nk.
We extended MCA by applying a vertex cover algorithm to compute Nk over a large (92,789 km2), continuous spatial scale for female bobcats (Lynx rufus) in Indiana, USA. We incorporated uncertainty by calculating confidence intervals for Nk across five thresholds of habitat suitability using 10 replicate suitability maps from bootstrapped datasets. For portions of the landscape too large to be solved with the vertex cover algorithm, we compared predictions from a linear model and a “greedy” algorithm.
Mean estimates of Nk for female bobcats in Indiana across habitat suitability thresholds ranged from 539 (0.75 threshold) to 1200 territories (0.25 threshold). On average, each 12.5 percentile reduction in the suitability threshold increased estimates for Nk by 1.2-fold. Both the predictive and greedy algorithm produced reasonable estimates of maximum cliques for areas that were too large to compute with the vertex cover algorithm. The greedy algorithm produced smaller confidence intervals compared to the predictive approach but underestimated maximum cliques by 1.2%.
Our research demonstrates effective application of MCA to species occupying large landscapes while accounting for uncertainty. We believe our methods, coupled with availability of annotated scripts developed in R, will make MCA more broadly accessible to wildlife biologists.
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Code and Data Availability
Code and data for maximum clique analysis are shared publicly in the Purdue University repository (https://purr.purdue.edu/publications/4058/1).
Akiba T, Iwata Y (2016) Branch-and-reduce exponential/FPT algorithms in practice: a case study of vertex cover. Theor Comput Sci 609:211–225
Aldridge CL, Boyce MS (2008) Accounting for fitness: combining survival and selection when assessing wildlife-habitat relationships. Isr J Ecol Evol 54:389–419
Andersen EM, Lovallo MJ (2003) Bobcat and Lynx. In: Feldhamer GA, Thompson BC, Chapman JA (eds) Wild mammals of North America: biology, management, and conservation. Johns Hopkins University Press, Baltimore, pp 758–786
Ayllón D, Almodóvar A, Nicola GG, Parra I, Elvira B (2012) Modelling carrying capacity dynamics for the conservation and management of territorial salmonids. Fish Res 134:95–103
Beissinger SR, McCullough DR (2002) Population viability analysis. University of Chicago Press, Chicago
Bolker BM (2008) Ecological models and data in R. Princeton University Press, Princeton
Brown ML, Donovan TM, Mickey RM, Warrington GS, Schwenk WS, Theobald DM (2018) Predicting effects of future development on a territorial forest songbird: methodology matters. Landsc Ecol 33:93–108
Burgman MA, Breininger DR, Duncan BW, Ferson S (2001) Setting reliability bounds on habitat suitability indices. Ecol Appl 11:70–78
Butenko S, Pardalos P, Sergienko I, Shylo V, Stetsyuk P (2009) Estimating the size of correcting codes using extremal graph problems. In: Pearce C, Hunt E (eds) Optimization. Springer, New York, pp 227–243
Catlin D, Gibson D, Friedrich MJ, Hunt KL, Karpanty SM, Fraser JD (2019) Habitat selection and potential fitness consequences of two early-successional species with differing life‐history strategies. Ecol Evol 9:13966–13978
Chapman EJ, Byron CJ (2018) The flexible application of carrying capacity in ecology. Glob Ecol Conserv 13:e00365
Clavero M, Hermoso V, Brotons L, Delibes M (2010) Natural, human and spatial constraints to expanding populations of otters in the Iberian Peninsula. J Biogeogr 37:2345–2357
Clutton-Brock TH, Coulson TN, Milner-Gulland EJ, Thomson D, Armstrong HM (2002) Sex differences in emigration and mortality affect optimal management of deer populations. Nature 415:633–637
Cochrane JC, Kirby JD, Jones IG, Conner LM, Warren RJ (2006) Spatial organization of adult bobcats in a longleaf pine-wiregrass ecosystem in southwestern Georgia. Southeast Nat 5:711–725
Cormen TH, Leiserson CE, Rivest RL, Stein C (2009) Introduction to algorithms. MIT Press, Cambridge
Csardi MG (2019) Package ‘igraph.’
Dalerum F, Cameron EZ, Kunkel K, Somers MJ (2009) Diversity and depletions in continental carnivore guilds: implications for prioritizing global carnivore conservation. Biol Lett 5:35–38
Del Monte-Luna P, Brook BW, Zetina‐Rejón MJ, Cruz‐Escalona VH (2004) The carrying capacity of ecosystems. Glob Ecol Biogeogr 13:485–495
Di Cola V, Broennimann O, Petitpierre B, Breiner FT, D'Amen M, Randin C, Engler R, Pottier J, Pio D, Dubuis A, Pellissier L, Mateo RG, Hordijk W, Salamin N, Guisan A (2017) ecospat: an R package to support spatial analyses and modeling of species niches and distributions. Ecography (Cop) 40:774–787
Diefenbach DR, Hansen LA, Warren RJ, Conroy MJ (2006) Spatial organization of a reintroduced population of bobcats. J Mammal 87:394–401
Donovan TM, Freeman M, Abouelezz H, Royar K, Howard A, Mickey R (2011) Quantifying home range habitat requirements for bobcats (Lynx rufus) in Vermont, USA. Biol Conserv 144:2799–2809
Donovan TM, Warrington GS, Schwenk WS, Dinitz JH (2012) Estimating landscape carrying capacity through maximum clique analysis. Ecol Appl 22:2265–2276
Elith J, Burgman MA, Regan HM (2002) Mapping epistemic uncertainties and vague concepts in predictions of species distribution. Ecol Modell 157:313–329
Feldhamer GA, Thompson BC, Chapman JA (2003) Wild mammals of North America: biology, management, and conservation. JHU Press, Baltimore
Ferguson AW, Currit NA, Weckerly FW (2009) Isometric scaling in home-range size of male and female bobcats (Lynx rufus). Can J Zool 87:1052–1060
Garey MR, Johnson DS (1979) Computers and intractability: a guide to the theory of NP-completedness. W. H. Freeman and Company, New York
Goss-Custard JD, Stillman RA, Caldow RWG, West AD, Guillemain M (2003) Carrying capacity in overwintering birds: when are spatial models needed? J Appl Ecol 40:176–187
Guisan A, Thuiller W, Zimmermann NE (2017) Habitat suitability and distribution models: with applications in R. Cambridge University Press, Cambridge
Hayward MW, O’Brien J, Kerley GIH (2007) Carrying capacity of large African predators: predictions and tests. Biol Conserv 139:219–229
Hilborn R, Walters CJ, Ludwig D (1995) Sustainable exploitation of renewable resources. Annu Rev Ecol Syst 26:45–67
Homer C, Dewitz J, Jin S, Xian G, Costello C, Danielson P, Gass L, Funk M, Wickham J, Stehman S, Auch R, Riitters K (2020) Conterminous United States land cover change patterns 2001–2016 from the 2016 national land cover database. ISPRS J Photogramm Remote Sens 162:184–199
Jacques CN, Klaver RW, Swearingen TC, Davis ED, Anderson CR, Jenks JA, Deperno CS, Bluett RD (2019) Estimating density and detection of bobcats in fragmented midwestern landscapes using spatial capture–recapture data from camera traps. Wildl Soc Bull 43:256–264
Johnson RR, Baxter CK, Estey ME (2009) An emerging agency-based approach to conserving populations through strategic habitat conservation. Models for planning wildlife conservation in large landscapes. Academic Press, Burlington, pp 01–223
Johnson SA, Walker HD, Hudson CM (2010) Dispersal characteristics of juvenile bobcats in south-central Indiana. J Wildl Manage 74:379–385
Jones LR, Zollner PA, Swihart RK, Godollei E, Hudson CM, Johnson SA (2020) Survival and mortality sources in a recovering population of bobcats (Lynx rufus) in south-central Indiana. Am Midl Nat 184:222–232
Jones LR, Johnson SA, Hudson CM, Zollner PA, Swihart RK (2022) Habitat selection in a recovering bobcat (Lynx rufus) population. PLOS One, in press
Larson MA, Thompson FR III, Millspaugh JJ, Dijak WD, Shifley SR (2004) Linking population viability, habitat suitability, and landscape simulation models for conservation planning. Ecol Modell 180:103–118
Leasure DR, Wenger SJ, Chelgren ND, Neville HM, Dauwalter DC, Bjork R, Fesenmyer KA, Dunham JB, Peacock MM, Luce CH, Lute AC, Isaak DJ (2019) Hierarchical multi-population viability analysis. Ecology 100:e02538
Li C (2019) JuliaCall: an R package for seamless integration between R and Julia. J Open Source Softw 4:1284
Losier CL, Couturier S, St-Laurent M, Drapeau P, Dussault C, Rudolph T, Brodeur V, Merkle JA, Fortin D (2015) Adjustments in habitat selection to changing availability induce fitness costs for a threatened ungulate. J Appl Ecol 52:496–504
Lyons AL, Gaines WL, Singleton PH, Kasworm WF, Proctor MF, Begley J (2018) Spatially explicit carrying capacity estimates to inform species specific recovery objectives: Grizzly bear (Ursus arctos) recovery in the North Cascades. Biol Conserv 222:21–32
MacKenzie DI, Nichols JD, Royle JA, Pollock KH, Bailey LL, Hines JE (2006) Occupancy estimation and modeling: inferring patterns and dynamics of species occurrence. Academic Press, San Diego
Mandujano S (2007) Carrying capacity and potential production of ungulates for human use in a Mexican tropical dry forest. Biotropica 39:519–524
Manly BFL, McDonald LL, Thomas DL, McDonald TL, Erickson WP (2002) Resource selection by animals: statistical design and analysis for field studies, 2nd edn. Kluwer Academic, Dordrecht
Maresh Nelson SB, Coon JJ, Miller JR (2020) Do habitat preferences improve fitness? Context-specific adaptive habitat selection by a grassland songbird. Oecologia 193:15–26
Martin BA, Shao G, Swihart RK, Parker GR, Tang L (2008) Implications of shared edge length between land cover types for landscape quality: the case of Midwestern US, 1940–1998. Landsc Ecol 23:391–402
McClure CJW, Westrip JRS, Johnson JA, Schulwitz SE, Virani MZ, Davies R, Symes A, Wheatley H, Thorstrom R, Amar A, Buij R, Jones VR, Williams NP, Buechley ER, Butchart SHM (2018) State of the world’s raptors: distributions, threats, and conservation recommendations. Biol Conserv 227:390–402
Millspaugh J, Thompson FR (2011) Models for planning wildlife conservation in large landscapes. Academic Press, London
Milner-Gulland EJ, Akçakaya HR (2001) Sustainability indices for exploited populations. Trends Ecol Evol 16:686–692
Mladenoff DJ, Sickley TA, Wydeven AP (1999) Predicting gray wolf landscape recolonization: logistic regression models vs. new field data. Ecol Appl 9:37–44
Morris DW, Mukherjee S (2007) Can we measure carrying capacity with foraging behavior? Ecology 88:597–604
Morrison ML (2001) A proposed research emphasis to overcome the limits of wildlife-habitat relationship studies. J Wildl Manage 65:613–623
Murphy DD, Noon BD (1991) Coping with uncertainty in wildlife biology. J Wildl Manage 55:773–782
Niskanen S, Östergård PRJ (2003) Cliquer user’s guide. version 1.0. Technical report
Preuss TS, Gehring TM (2007) Landscape analysis of bobcat habitat in the northern lower peninsula of Michigan. J Wildl Manage 71:2699–2706
R Development Core Team (2018) R: A language and environment for statistical computing
Ripple WJ, Estes JA, Beschta RL, Wilmers CC, Ritchie EG, Hebblewhite M, Berger J, Elmhagen B, Letnic M, Nelson MP, Schmitz OJ, Smith DS, Wallach AD, Wirsing AJ (2014) Status and ecological effects of the world’s largest carnivores. Sci (80) 343:1241484
Tinker MT, Yee JL, Laidre KL, Hatfield BB, Harris MD, Tomoleoni JA, Bell TW, Saarman E, Carswell LP, Miles AK (2021) Habitat features predict carrying capacity of a recovering marine carnivore. J Wildl Manage 85:303–323
Tucker SA, Clark WR, Gosselink TE (2008) Space use and habitat selection by bobcats in the fragmented landscape of south-central Iowa. J Wildl Manage 72:1114–1124
Urbanek S, Urbanek MS (2016) Package ‘rJava’
Williams BK, Nichols JD, Conroy MJ (2002) Analysis and management of animal populations. Academic Press, London
Woodworth-Jefcoats PA, Polovina JJ, Drazen JC (2017) Climate change is projected to reduce carrying capacity and redistribute species richness in North Pacific pelagic marine ecosystems. Glob Chang Biol 23:1000–1008
Woolf A, Nielsen CK, Weber T, Gibbs-Kieninger TJ (2002) Statewide modeling of bobcat, Lynx rufus, habitat in Illinois, USA. Biol Conserv 104:191–198
Yokomizo H, Possingham HP, Thomas MB, Buckley YM (2009) Managing the impact of invasive species: the value of knowing the density–impact curve. Ecol Appl 19:376–386
We thank all the volunteers, licensed trappers, deer archery hunters, Purdue University deer project technicians, DNR Furbearer program technicians, and Indiana Conservation Officers who helped collect bobcat location data. We thank Jacob Peterson for assistance with spatial analyses. We gratefully acknowledge the traditional homelands of the Indigenous People upon which Purdue University is built and our fieldwork was conducted. We honor and appreciate the Bodéwadmik (Potawatomi), Lenape (Delaware), Myaamia (Miami), and Shawnee People who are the original Indigenous caretakers.
This study was funded by Indiana Department of Natural Resources’ State Wildlife grant T3S series and Wildlife Restoration Grant W45R3, USDA National Institute of Food and Agriculture McIntire Stennis Project #1010322, Purdue University Department of Forestry and Natural Resources, and public donations to the Indiana Nongame Wildlife Fund.
The authors declare that they have no competing interests.
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Jones, L.R., Swihart, R.K., Gleich, D.F. et al. Estimating statewide carrying capacity of bobcats (Lynx rufus) using improved maximum clique algorithms. Landsc Ecol 37, 2383–2397 (2022). https://doi.org/10.1007/s10980-022-01460-6
- Ensembles of small models
- Habitat suitability
- Home range capacity
- Landscape carrying capacity
- Maximum clique analysis
- N k