lec7 05

Slide1: 

Predation All organisms are subject to various sources of mortality (starvation disease, physical injury, and predation) Understanding how much ‘natural’ mortality populations receive is critical to managing populations that we exploit (e.g. fisheries) Conceptual models of predation: Process of predation can be broken into a series of steps: -searching -catching -subduing -consuming -digesting Each of these steps takes time, and these times vary

Slide2: 

Lotka-Volterra predator-prey model (Gotelli 2001 Primer in Ecology): First consider the prey: Prey in the absence of predators: dV/dt = rV Where V = prey (victim) population, r = intrinsic rate of increase Prey in the presence of predators dV/dt = rV - VP Where VP is the loss to predators Loss to predators is determined by the product of predator and victim numbers (assumes predators and prey move randomly through the environment) and the capture efficiency,   is a measure of the effect of the predator on the per capita growth rate of prey

Slide3: 

Larger : more the per capita growth rate of the prey population is depressed by the addition of a single predator Here V also represents the functional response of the predator (this describes how the rate of prey capture is affected by prey abundance). In this case, the functional response is linear - capture rate increases at a constant rate as prey density increases. Now for predator population growth: In the simplest form of the model, the predator is specialized on just one prey species - therefore in the absence of prey, the predator population declines exponentially: dP/dt = -sP where P is the predator population size, and s is the per capita death rate

Slide4: 

Positive population growth occurs when prey are present as follows: dP/dt = ßVP - sP Where ß is the conversion efficiency - the ability of predators to turn prey into additional per capita growth rate for the predator population. When ß is high a single prey item is valuable (killing a Woolly Mammoth…). ßV is called the numerical response of the predator population - the per capita growth rate of the predator population as a function of the prey population.

Slide5: 

Equilibrium solutions: (starting with the prey population) dV/dt = rV - VP 0 = rV - VP rV = VP r = P Equilibrium P = r/ A specific number of predators will keep the prey population at zero growth. How many depends on the ratio of the growth rate of prey to the capture efficiency of the predators. For the predator population: dP/dt = ßVP - sP 0 = ßVP - sP ßVP = sP ßV = s Equilibrium V = s/ß depends on the ratio of the death rate of predators to the conversion efficiency of predators

Slide6: 

Prey (victim) zero growth isocline

Slide7: 

Predator zero growth isocline (notice axes haven’t changed…)

Slide8: 

P abundant V scarce P&V decrease P scarce V scarce V recovers P scarce P recovers V increases P&V abundant P increases V decreases V isocline P isocline

Slide9: 

Unless P and V are at the intersect of zero growth isoclines or (P>>V) populations will cycle (neutral stability). Frequency depends on r and s

Slide10: 

Assumptions of the simple model: 1. Prey population growth is only limited by predators (no effect of K) 2. Predator is a specialist unable to switch among prey species 3. Prey zero growth isocline is horizontal - therefore predators can consume an infinite number of prey (doesn’t matter how many prey are present, one more predator is enough to drop prey population growth below zero 4. Predators and prey encounter each other at random - no refuges from predators

Slide11: 

Getting real… Add a prey carrying capacity dV/dt = rV -VP -cV2 Where c is a constant Now prey population growth will slow with increasing abundance even in the absence of predators Result: Prey zero-growth isocline will have a negative slope

Slide12: 

Prey carrying capacity results in damped oscillations leading to equilibrium predator and prey populations

Slide13: 

Making prey capture and consumption more realistic • Existing model - predators increase prey consumption at a constant rate as prey abundance increases. • Need to incorporate a non-linear functional response of predators to prey Holling’s model: consider time needed to “handle” each prey (capture and consumption) (= th), and time to search for prey (=ts). Handling time (th), further broken down to: h = per prey handling time n = number of prey captured in time t (capture rate) th = hn

Slide14: 

Capture rate (n) in turn depends prey population size (V), capture efficiency (), and time spent searching (ts). n = Vts so… ts = n/V So total feeding time (t) = ts + th = n/V + hn Gives us type II functional response n/t = V(1+ Vh) Type II response: predators can be satiated so that prey captured per predator per unit time reaches an asymptote

Slide15: 

High prey densities n/t determined by h

Slide16: 

Holling’s Type II Functional response fits data on fish predation quite well... 3 different predators, 4 different predator sizes Single prey species present at constant initial concentration

Slide17: 

Holling’s Type II functional response is equivalent to Michaelis - Menton equation describing enzyme kinetics Sometimes a logistic function better describes predator functional response (low consumption rate at low prey abundance). Examples of that?

Slide18: 

Type II or III without carrying capacity are not stable

Slide19: 

Most appropriate model yields humped prey isocline?

Slide20: 

Inefficient predator: populations oscillate to equilibrium point

Slide21: 

Efficient predator: predator drives prey to extinction (“paradox of enrichment”)