Predator-prey equations

Readings Predator-prey: Chapter 12: Case TJ (2000) An illustrated guide to theoretical ecology. Oxford University Press, Oxford. Multispecies exploitation: Branch TA et al. (2013) Opportunistic exploitation: an overlooked pathway to extinction. Trends in Ecology and Evolution. doi: /j.tree

The predators and prey

Simple predator-prey theory (Lotka-Volterra) Prey governed by exponential growth Predator deaths are density independent, births depend upon number of prey eaten Prey eaten per predator is proportional to prey density Lotka AJ (1925) Elements of physical biology. Williams & Wilkins Co., Baltimore Volterra V (1926) Variazioni e fluttuazioni del numero d'individui in specie animali conviventi. Mem. Acad. Lincei Roma 2:31-113

Lotka-Volterra equations Natural mortality Wildebeest numbers Lion numbers Predation efficiency Intrinsic rate of increase Assimilation efficiency Equivalent in time steps

Dynamic behavior These models are either unstable or cyclic

Biological unrealism of Lotka-Volterra No prey self limitation No predator self limitation No limit on prey consumption per predator – This is called the “functional response”

Adding some biological realism Survival Proportion of prey searched for, found and killed per year per predator Number of wildebeest deaths Assimilation: number of lions produced per wildebeest death Logistic equation “Functional response” Lab 5 Predator prey wildebeest lions.xlsx

Dynamic behavior in time Wildebeest numbers Lion numbers Wildebeest numbers Lion numbers Lab 5 Predator prey wildebeest lions.xlsx

Predator-prey phase diagram Lab 5 Predator prey wildebeest lions.xlsx

Predation dynamics Developing a functional response Predators do a random walk, encounter and kill a fraction of what prey they encounter. Exponential model is used to correct for the fact that no prey can be killed and eaten twice by different lions. Wildebeest deaths Proportion killed by one lion Lab 5 Predator prey wildebeest lions.xlsx

Lion kill rate (one way to think about estimating h) A lion walks 10 km per day Can see 200 m in either direction Thus sees all wildebeest covering 4 km 2 /day This amounts to 1460 km 2 /year Serengeti ecosystem is 90,000 km 2 A lion can chase and catch 1 in 1000 animals it sees Thus one lion kills h = (1460/90,000)/1000 = of the wildebeest population per year

Key assumptions Kill rate proportional to prey abundance No self regulation of predator No predator saturation

The prey isocline When is prey abundance constant? Original equation Equilibrium implies W t+1 = W t = W Divide by W Rearrange Solve for W Lab 5 Predator prey wildebeest lions.xlsx

The predator isocline When is predator abundance constant? Original equation Set L t+1 = L t = L Divide by L Solve for W Lab 5 Predator prey wildebeest lions.xlsx

Predator isocline Prey isocline Predator increasing Prey declining Predator declining Prey declining Predator declining Prey increasing Predator increasing Prey increasing Lab 5 Predator prey wildebeest lions.xlsx

Multiple prey species

Hyper- predation Channel Islands: introduced feral pigs allowed golden eagles to establish and increase, greatly increasing predation (hyperpredation) on native foxes. Low fox numbers allowed competitively inferior skunks to flourish. Multispecies equations No pigsPigs Roemer GW et al. (2002) Golden eagles, feral pigs, and insular carnivores: How exotic species turn native predators into prey. PNAS 99:

Opportunistic exploitation S.H. pelagic whaling: despite rarity of blue whales, whaling continued on fin whales; whalers opportunistically caught valuable blue whales when encountered Branch TA et al. (2013) Opportunistic exploitation: an overlooked pathway to extinction. Trends in Ecology and Evolution. doi: /j.tree

Opportunistic exploitation Branch TA et al. (2013) Opportunistic exploitation: an overlooked pathway to extinction. Trends in Ecology and Evolution. doi: /j.tree