EXPLOITATION + Species 2 (predator P) Species 1 (victim V) - Classic predation theory is built upon the idea of time constraint (foraging theory): A 24 hour day is divided into time spent unrelated to eating: social interactions mating rituals grooming sleeping And eating-related activities: searching for prey pursuing prey subduing the prey eating the prey digesting (may not always exclude other activities) The time constraints on foraging other essential activities foraging Foraging time The time constraints on foraging Foraging time other essential activities Handling time search handling Search time Search time: all activities up to the point of spotting the prey searching Handling time: all activities from spotting to digesting the prey pursuing subduing, killing eating (transporting, burying, regurgitating, etc) digesting Caveat: not all activities may be mutually exclusive ex. Digesting and non-eating related activities The time constraints on foraging Foraging time other essential activities Handling time eating pursuing & subduing Search time search eating pursuit & time subdue time Different species will allocate foraging time differently: Filter feeder: Sit & wait predator (spider) eating waiting eating digesting subduing Time allocation also depends on victim density and predator status: Well-fed mammalian predator: Starving mammalian Predator (victims at low dnsity): eating eating pursuing & subduing searching pursuing searching & subduing The math of predation: (After C.S. Holling) ts th Total search time per day Total handing time per day tmax ts th C.S. (Buzz) Holling Total foraging time is fixed (or cannot exceed a certain limit). 1) Define the per-predator capture rate as the number of victims captured (n) per time spent searching (ts): n ts 2) Capture rate is a function of victim density (V). Define a as capture efficiency. n aV ts 3) Every captured victim requires a certain time for “processing”. th hn t ts th n ts aV th hn n t hn aV n aV t 1 ahV n/t = capture rate n aV t 1 ahV 0.1 Capture rate 0.09 0.08 0.07 0.06 0.05 0.04 Capture rate limited by prey density and capture efficiency 0.03 0.02 0.01 0 0 50 Capture rate limited by predator’s handling time. 100 Prey density (V) 150 Damselfly nymph (Thompson 1975) Asymptote: 1/h Decreasing prey size The larger the prey, the greater the handling time. (Thompson 1975) Three Functional Responses (of predators with respect to prey abundance): Holling Type I: Consumption per predator depends only on capture efficiency: no handling time constraint. Holling Type II: Predator is constrained by handling time. Holling Type III: Predator is constrained by handling time but also changes foraging behavior when victim density is low. Per predator consumption rate Type I (filter feeders) Type II (predator with significant handling time limitations) Type I: n aV t Type II n aV t 1 ahV Type III (predator who pays less attention to victims at low density) victim density Type III n (aV ) 2 t 1 (ahV ) 2 Holling Type I functional response: Type I functional response Daphnia path Daphnia (Filter feeder on microscopic freshwater organism) Thin algae suspension culture Thick algae suspension culture Holling Type II functional response: Slug eating grass Cattle grazing in sagebrush grassland Holling Type III functional response: Paper wasp, a generalist predator, eating shield beetle larvae: The wasp learns to hunt for other prey, when the beetle larvae becomes scarce. The dynamics of predator prey systems are often quite complex and dependent on foraging mechanics and constraints. Gause’s Predation Experiments: Didinium nasutum eats Paramecium caudatum: Gause’s Predation Experiments: 1) Paramecium in oat medium: logistic growth. 2) Paramecium with Didinium in oat medium: extinction of both. 3) Paramecium with Didinium in oat medium with sediment: extinction of Didinium. A fly and its wasp predator: Greenhouse whitefly Laboratory experiment Parasitoid wasp (Burnett 1959) spider mite on its own with predator in simple habitat Spider mites with predator in complex habitat (Laboratory experiment) Predatory mite (Huffaker 1958) Azuki bean weevil and parasitoid wasp (Laboratory experiment) (Utida 1957) stoat (Greenland) collared lemming lemming stoat (Gilg et al. 2003) Possible outcomes of predator-prey interactions: 1. The predator goes extinct. 2. Both species go extinct. 3. Predator and prey cycle: prey boom predator boom Predator bust prey bust 4. Predator and prey coexist in stable ratios. Putting together the population dynamics: Predators (P): dP dt Victim consumption rate * Victim Predator conversion efficiency - Predator death rate Victims (V): dV dt Victim renewal rate – Victim consumption rate Choices, choices…. Victim growth assumption: • exponential • logistic Functional response of the predator: •always proportional to victim density (Holling Type I) •Saturating (Holling Type II) •Saturating with threshold effects (Holling Type III) The simplest predator-prey model (Lotka-Volterra predation model) dV rV aVP dt dP VP qP dt Exponential victim growth in the absence of predators. Capture rate proportional to victim density (Holling Type I). Isocline analysis: dV r 0: P dt a dP q 0 :V dt Victim isocline: Predator isocline: Predator density V q Victim density P r a V Predator density dV/dt < 0 dP/dt < 0 q dV/dt < 0 dP/dt > 0 dV/dt > 0 dP/dt < 0 Predator isocline: Victim isocline: dV/dt > 0 dP/dt > 0 Victim density Show me dynamics P r a Victim isocline: Predator isocline: Predator density V q Victim density P r a Victim isocline: Preator isocline: Predator density V q Victim density P r a Neutrally stable cycles! Every new starting point has its own cycle, except the equilibrium point. The equilibrium is also neutrally stable. Victim isocline: Preator isocline: Predator density V q Victim density P r a dV V rV 1 aVP dt K dP VP qP dt Logistic victim growth in the absence of predators. Capture rate proportional to victim density (Holling Type I). Predator isocline: Predator density r a Victim density r c Stable Point ! P Predator and Prey cycle move towards the equilibrium with damping oscillations. V dV aVP rV dt V D dP VP qP dt V D Exponential growth in the absence of predators. Capture rate Holling Type II (victim saturation). Predator density Predator isocline: r kD Victim density Unstable Equilibrium Point! Predator and prey move away from equilibrium with growing oscillations. P V No density-dependence in either victim or prey (unrealistic model, but shows the propensity of PP systems to cycle): P V Intraspecific competition in prey: (prey competition stabilizes PP dynamics) P V Intraspecific mutualism in prey (through a type II functional response): P V Predators population growth rate (with type II funct. resp.): dP VP qP dt V D Victim population growth rate (with type II funct. resp.): dV V aVP rV 1 dt K V D Predator isocline: Predator density Rosenzweig-MacArthur Model Victim density If the predator needs high victim density to survive, competition between victims is strong, stabilizing the equilibrium! Predator isocline: Predator density Rosenzweig-MacArthur Model Victim density If the predator drives the victim population to very low density, the equilibrium is unstable because of strong mutualistic victim interactions. Predator isocline: Predator density Rosenzweig-MacArthur Model Victim density However, there is a stable PP cycle. Predator and prey still coexist! Predator isocline: Predator density Rosenzweig-MacArthur Model Victim density The Rosenzweig-MacArthur Model illustrates how the variety of outcomes in Predator-Prey systems can come about: 1) Both predator and prey can go extinct if the predator is too efficient capturing prey (or the prey is too good at getting away). 2) The predator can go extinct while the prey survives, if the predator is not efficient enough: even with the prey is at carrying capacity, the predator cannot capture enough prey to persist. 3) With the capture efficiency in balance, predator and prey can coexist. a) coexistence without cyclical dynamics, if the predator is relatively inefficient and prey remains close to carrying capacity. b) coexistence with predator-prey cycles, if the predators are more efficient and regularly bring victim densities down below the level that predators need to maintain their population size.