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## Predation and Competition

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**Predation and Competition**Abdessamad Tridane MTBI summer 2008**Types**Response of Sp A Response of Sp B Competition - - Predation + - Parasitism + - Parasitoidism + - Herbivory + - Neutral Mutualism Commensalism Amensalism Two-species interactions**Types of predators**Carnivores – kill the prey during attack Herbivores–remove parts of many prey, rarely lethal. Parasites – consume parts of one or few prey, rarely lethal. Parasitoids –kill one prey during prolonged attack.**How has predation influenced evolution?**Adaptations to avoid being eaten: spines (cactii, porcupines) hard shells (clams, turtles) toxins (milkweeds, some newts) bad taste (monarch butterflies) camouflage aposematiccolors mimicry**Mimicry– look like something that is dangerous or tastes**bad**Mimicry– look like something that is dangerous**or tastes bad Mullerian mimicry – convergence of several unpalatable species**Mimicry– look like something that is dangerous**or tastes bad Batesian mimicry – palatable species mimics an unpalatable species model mimic mimics model**A verbal model of predator-prey cycles:**• Predators eat prey and reduce their numbers • Predators go hungry and decline in number • With fewer predators, prey survive better and increase • Increasing prey populations allow predators to increase • And repeat…**The Lotka-Volterra Model: Assumptions**Prey grow exponentially in the absence of predators. Predation is directly proportional to the product of prey and predator abundances (random encounters). Predator populations grow based on the number of prey. Death rates are independent of prey abundance.**An introduction to prey-predator Models**• Lotka-Volterra model • Lotka-Volterra model with prey logistic growth • Holling type II model**Generic Model**• f(x) prey growth term • g(y) predator mortality term • h(x,y) predation term • e prey into predator biomass conversion coefficient**Lotka-Volterra Model**• r prey growth rate : Malthus law • m predator mortality rate : natural mortality • Mass action law • a and b predation coefficients : b=ea • e prey into predator biomass conversion coefficient**Number of predators depends on the prey population.**Predator isocline Number of Predators (y) Predators decrease Predators increase m/b Number of prey (x)**Number of prey depends on the predator population.**Prey decrease Prey Isocline Number of Predators (y) r/a Prey increase m/b Number of prey (x)**Local stability analysis**• Jacobian at positive equilibrium • detJ*>0 and trJ*=0 (center)**Local stability analysis**• Proof of existence of center trajectories (linearization theorem) • Existence of a first integral H(x,y) :**Freshmen and donuts: an example**• There is a room with 100 donuts – what does a typical male freshmen do? • First – eat several donuts. (A male freshman can eat 10 donuts) • Second – rapidly tell friends • But not too many! • Third – Room reaches carrying capacity at 10 male freshmen. • So K=10 for male freshmen.**Nullclines for the Lotka-Volterra model with prey logistic**growth**Lotka-Volterra Model with prey logistic growth**• Equilibrium points : (0,0) (K,0) (x*,y*)**Local stability analysis**• Jacobian at positive equilibrium • detJ*>0 and trJ*<0 (stable)**Lotka-Volterra model with prey logistic growth : coexistence****Lotka-Volterra with prey logistic growth : predator**extinction**Transcritical bifurcation**(K,0) stable and (x*,y*) unstable and negative (K,0) and (x*,y*) same (K,0) unstable and (x*,y*) stable and positive**Loss of periodic solutions**coexistence Predator extinction**How do species interact?**• Competition • Predation • Herbivory • Parasitism • Disease • Mutualism**Interspecific Competition**• Competition • When two species use the same limited resource to the detriment of both species. • Assessment-some general features of interspecific competition • Competitive exclusion or coexistence • Tilman’s model of competition for specific resources (ZINGIs) • Coexistence: reducing competition by dividing resources**Assessment**• mechanisms • consumptive or exploitative — using resources (most common) • preemptive — using space, based on presence • overgrowth — exploitative PLUS preemptive • chemical — antibiotics or allelopathy • territorial — like preemptive, but behavior • encounter — chance interactions**Modeling coexistence?**• Can we model the growth of 2 species? • Remember logistic model? • What is K? • Now we add another factor that can limit the abundance of a species. • Another species.**Freshmen and donuts: an example**• What happens if a male and female discover the room at the same time? • First – eat several donuts. (A female freshman can eat 5 donuts) • Second – rapidly tell friends • But not too many! • Third – Room reaches carrying capacity at ? males and ? females. • What is the carrying capacity? • It depends…**Lotka-Volterra**• Need a way to combine the two equations. • If species are competing, the number of species A decreases if number of species B increases. • Such that: • Where alpha is the competition coefficient • Lotka-Volterra: A logistic model of interspecific competition of intuitive factors.**Freshman Example**• In a room we have 100 donuts. • Need 10 donuts for each male freshmen. • So K1 = 10 • Need only 5 donuts for each female freshmen. • So K2 = 20 • If room is at K1 and 1 male leaves, how many females can come in? • So, , where α = 0.5 • And, , where B = 2**Possible outcomes when put two species together.**• Species A excludes Species B • Species B excludes Species A • Coexistence