C. Models

1. Pathogens

C. Models

1. Pathogens

R = (b/g)S

b = rate of transmissiong = recovery time (inverse of infectious period)

C. Models

2. Lotka-Volterra Models

Goal - create a model system in which there are oscillations of predator and prey populations that are out-of-phase with one another. Basic Equations:

a. Prey Equation: dV/dt = rV - cVP where

rV defines the maximal, geometric rate c = predator foraging efficiency: % eaten P = number of predators V= number of prey, so PV = number

of encounters and cPV = number of prey killed (consumed) So, the formula describes the maximal growth rate, minus the number of prey individuals lost by predation.

 

  

Number of Prey (V)

C. Models

2. Lotka-Volterra Models

b. Predator The Equation: dP/dt = a(cPV) - dP where

CPV equals the number of prey consumed, and a = the rate at which food energy is converted to offspring. So, a(cVP) = number of predator offspring produced. d = mortality rate, and P = # of predators, so dP = number of

carnivores dying. So, the equation boils down to the birth rate (determined by

energy "in" and conversion rate to offspring) minus the death rate.

 

  

Basic Equations:1. Prey 2. Predator3. Dynamics

 

  

1.

2. 3.

4.

1. 2. 3. 4. 1.

V. Dynamics of Consumer-Resource InteractionsA. Predators can limit the growth of prey populationsB. Oscillations are a Common PatternC. ModelsD. Lab Experiments

1. Gause

P. caudatum (prey) and Didinium nasutum (predator)

P. caudatum (prey) and Didinium nasutum (predator)

In initial experiments, Paramecium populations would increase, followed by a pulse of Didinium, and then they would crash.

 

  

P. caudatum (prey) and Didinium nasutum (predator)

In initial experiments, Paramecium populations would increase, followed by a pulse of Didinium, and then they would crash.

He added glass wool to the bottom, creating a REFUGE that the predator did not enter.

 

  

He induced oscillations by adding Paramecium as 'immigrants'

 

  

D. Laboratory Experiments

1. Gause2. Huffaker six-spotted mite (Eotetranychus sexmaculatus) was prey - SSM Predatory mite (Typhlodromus occidentalis) was predator - PM

 

  

 

  

D. Laboratory Experiments

1. Gause2. Huffaker 3. Holyoak and Lawler-1996

 

  

Holyoak and Lawler-1996 Used a bacteriovore ciliate, Colpidium striatum as the prey and our old friend Didinium nasutum as the predator.

 

  

3. Holyoak and Lawler-1996 Set up replicate 30mL bottles, linked together by tubes, and single flask systems.

 

  

D. Complexities and Applications

2. Multiple State StatesConsider a Type III functional response, where the predation rate is

highest at intermediate prey densities.

 

  

V

Birth rate

Predation Rate

D. Complexities and Applications

2. Multiple State StatesConsider a Type III functional response, where the predation rate is

highest at intermediate prey densities.

 

  

V

Birth rate

Predation Rate