Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
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: 10.1016/j.tree.2013.03.003 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 Intrinsic rate of increase Wildebeest numbers Lion numbers dW rW eWL dt Predation efficiency dL mL eaWL dt Natural mortality Assimilation efficiency Equivalent in time steps Wt 1 Wt rWt eWt Lt Lt 1 Lt mLt eaWt Lt Lt 1 sLt eaWLt Dynamic behavior These models are either unstable or cyclic 90,000,000 10,000,000 80,000,000 9,000,000 Wildebeest 70,000,000 8,000,000 Lions 60,000,000 7,000,000 6,000,000 50,000,000 5,000,000 40,000,000 4,000,000 30,000,000 3,000,000 20,000,000 2,000,000 10,000,000 1,000,000 0 0 0 50 150 100 Time 200 250 300 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 Logistic equation Survival Number of wildebeest deaths Wt Wt 1 Wt rWt 1 Dt K Assimilation: number of lions Lt 1 sLt aDt produced per wildebeest death Dt Wt 1 exp(hLt ) “Functional response” Proportion of prey searched for, found and killed per year per predator Lab 5 Predator prey wildebeest lions.xlsx Dynamic behavior in time 18,000 16,000 1,000,000 14,000 800,000 12,000 Wildebeest numbers 10,000 600,000 8,000 400,000 6,000 Lion numbers Lion numbers Wildebeest numbers 1,200,000 4,000 200,000 2,000 0 0 0 50 100 150 200 250 300 Lab 5 Predator prey wildebeest lions.xlsx Predator-prey phase diagram 20,000 Lions 15,000 10,000 5,000 0 0 500,000 1,000,000 1,500,000 Wildebeest Lab 5 Predator prey wildebeest lions.xlsx Predation dynamics Developing a functional response Wildebeest deaths Dt Wt 1 exp(hLt ) Fraction wildebeest killed • 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. Proportion killed by one lion 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 50000 100000 150000 200000 Number of lions 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 km2/day This amounts to 1460 km2/year Serengeti ecosystem is 90,000 km2 A lion can chase and catch 1 in 1000 animals it sees Thus one lion kills h = (1460/90,000)/1000 = 0.000016 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 Wt+1 = Wt = W Divide by W Rearrange Solve for W Wt Wt 1 Wt rWt 1 Wt 1 exp hLt K W W W rW 1 W 1 exp hLt K W 1 1 r 1 1 exp hLt K W r 1 1 exp hLt K K W r 1 exp hL r Lab 5 Predator prey wildebeest lions.xlsx The predator isocline When is predator abundance constant? Original equation Set Lt+1 = Lt = L Divide by L Solve for W Lt 1 Lt s aWt 1 exp hLt L Ls aW 1 exp hL a 1 s W 1 exp hL L 1 s L W a 1 exp hL Lab 5 Predator prey wildebeest lions.xlsx Predator isocline 20,000 Lions 15,000 10,000 5,000 Prey isocline 0 0 500,000 1,000,000 1,500,000 Wildebeest Lab 5 Predator prey wildebeest lions.xlsx Multiple prey species Multispecies equations Hyperpredation 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. No pigs Pigs Roemer GW et al. (2002) Golden eagles, feral pigs, and insular carnivores: How exotic species turn native predators into prey. PNAS 99:791-796 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: 10.1016/j.tree.2013.03.003 Opportunistic exploitation Branch TA et al. (2013) Opportunistic exploitation: an overlooked pathway to extinction. Trends in Ecology and Evolution. doi: 10.1016/j.tree.2013.03.003