Predation and Competition Abdessamad Tridane MTBI summer 2008 MTBI summer 2008

Two-species interactions Neutral Mutualism - Competition Amensalism Commensalism + Herbivory Parasitoidism Parasitism Predation Response of Sp B of Sp A Types

Predation

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, Parasitoids – kill one prey during prolonged attack.

Adaptations to avoid being eaten: 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 aposematic colors mimicry

Camouflage – blending in

Aposematic colors – warning

Mimicry – look like something that is dangerous or tastes bad MTBI summer 2008

Mimicry – look like something that is dangerous or tastes bad Mullerian mimicry – convergence of several unpalatable species MTBI summer 2008

Mimicry – look like something that is dangerous or tastes bad Batesian mimicry – palatable species mimics an unpalatable species model mimic mimics model MTBI summer 2008

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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… MTBI summer 2008

Why don’t predators increase at the same time as the prey? MTBI summer 2008

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. MTBI summer 2008

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 MTBI summer 2008

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 MTBI summer 2008

Number of predators depends on the prey population. isocline Number of Predators (y)‏ Predators decrease Predators increase m/b Number of prey (x)‏ MTBI summer 2008

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)‏ MTBI summer 2008

Lotka-Volterra nullclines MTBI summer 2008

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Direction field for Lotka-Volterra model MTBI summer 2008

Local stability analysis Jacobian at positive equilibrium detJ*>0 and trJ*=0 (center)‏ MTBI summer 2008

Linear 2D systems (hyperbolic)‏ MTBI summer 2008

Local stability analysis Proof of existence of center trajectories (linearization theorem)‏ Existence of a first integral H(x,y) : MTBI summer 2008

Lotka-Volterra model MTBI summer 2008

Lotka-Volterra model MTBI summer 2008

Hare-Lynx data (Canada)‏ MTBI summer 2008

Logistic growth (sheep in Australia)‏ MTBI summer 2008

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.

Lotka-Volterra Model with prey logistic growth MTBI summer 2008

Nullclines for the Lotka-Volterra model with prey logistic growth MTBI summer 2008

Lotka-Volterra Model with prey logistic growth Equilibrium points : (0,0) (K,0) (x*,y*)‏ MTBI summer 2008

Local stability analysis Jacobian at positive equilibrium detJ*>0 and trJ*<0 (stable)‏ MTBI summer 2008

Condition for local asymptotic stability MTBI summer 2008

Lotka-Volterra model with prey logistic growth : coexistence MTBI summer 2008

Lotka-Volterra with prey logistic growth : predator extinction MTBI summer 2008

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 MTBI summer 2008

Loss of periodic solutions coexistence Predator extinction MTBI summer 2008

Competition MTBI summer 2008

How do species interact? Competition Predation Herbivory Parasitism Disease Mutualism MTBI summer 2008

Interspecific 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 MTBI summer 2008

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 MTBI summer 2008

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. MTBI summer 2008

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… MTBI summer 2008

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. MTBI summer 2008

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 MTBI summer 2008

Possible outcomes when put two species together. Species A excludes Species B Species B excludes Species A Coexistence MTBI summer 2008

Changes in population 1: Yellow: both increase White: both decrease MTBI summer 2008

Changes in population 2: Yellow: both increase White: both decrease MTBI summer 2008

Yellow: both increase White: both decrease Green: Sp 1 increase Brown: Sp 2 increase MTBI summer 2008

Tilman’s model Problems with Lotka-Voltera model? No mechanism Logistic-competition theory is based on the dynamics of the consumer populations involved, i.e., it does not explicitly consider changes in resources utilized by the competitors. Tilman (1982) treated the regulation of population size from the standpoint of resource dynamics, i.e., supply and consumption. MTBI summer 2008

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1 – no species can survive 2 – Only A can live 3 – Species A out competes B 4 – Stable coexistence 5 – Species B out competes A 6 – Only B can live MTBI summer 2008

MTBI summer 2008

Will be posted on my website http://math.asu.edu/~tridane Homework Will be posted on my website http://math.asu.edu/~tridane