x             M

___     =    ___

n              N

Mark-Recapture method.

Used to estmate popn size.

x=marked individuals in the sample

n=sample size

M=total marked in the popn

N=total popn

So, N=nM

          ___

           x

N(t)=N(0)e^rt

N(t)= number of individuals in a popn after t time

N(0) = initial popn size

r= exponential growth rate

N(t+1)= N(t) lambda

lambda = ratio of popn size in one year compared to the next

Exponential gorwth:

constant growth in popn size over time.  r= rate of exponential growth.

Geometric Growth:

Growth that occurs in discrete intervals, annual growth rates.  lambda= rate of popn growth within a given time interval (ratio of popn size in one year to the next)

t2 =         log e 2

                --------------

                 log e lambda

Predicts oscillations in the abundance of P and R popns.

dR              

------ = rR - cRP

 dt             .  

[rate of change in the prey popn] = [growth rate of R] - removal of R by P]

two components:

1. unrestricted exponential growth in absense of predators rR

2. removal of prey by predators beyond other causes of death.

dP             .

---- = acRP-dP

dt              .

Two components:

1.  Birth rate which depends on # R captured. (acRP)

2.  Death rate from outside system. (dP)

a = efficiency with which food is converted to popn growth.

d = death

d = change

Predator achieves equilibrium

dR     .

----- = 0

dt     .

 change in prey popn divided by change in time is 0

So, rR=cRP

rate of increase in prey popn equals removal of prey individuals by predators (c, capture efficiency of prey by predators).

P*=  r

        ----

       c

dP    .

---- = 0

dt    .

rate of predator change over change in time is 0.

So, acRP = dP

birth term of predators (acRP -> # prey captured cRP times coefficient , a, for efficiency with which food is converted to popn growth) equals death term of predators.

R*=  d

        ---

         ac 

they depress R numbers slightly but R popn remains near equilibrium level set by its resources.

page 361, fig 18.18a

predator control of R popn.

page 361, 18.18b

Predation may effectively limit R popn growth under all circumstances and equilibrium point C disapears.

page 361, fig 18.18C

that all R are eaten to extinction.

page 361, fig 18.18D

Describes the relationship between rate at which P consumes R and R density.

Relationship of an individual predator's rate of food consumption to the density of prey.

1.Density-dependent limitation of either predator or prey by external factors (eg prey resource limitation).
2.Predator inefficiency or enhanced prey escape or defence (lead to lower cvalues—leads to P↑R↑)
3.Alternative food sources that the predators switch to when the main prey is scarce.
4.Refuges from predation at low prey densities.
5.Reduced time delays in predator response to changes in prey abundance.
6.Spatial patchiness—Huffaker experiment.

1. resources that have a source external to the system, beyond influence of the consumer

2.  Resources generated within the ecosystem and their abundance are directly depressed by consumers

3.  Resources regenerated within the ecosystem but resource and consumer are linked indirectly. eg nutrient cycles

No two organisms can coexist exclusively on the same limiting resources.

Accomplished by direct physical interaction

1    dN     r[K-N/K]

  ----  ----  =               .

N     dt                .

 Popn continues to increase until N=K

Growth rate is depressed by intraspecific competition.

1/N₁ * dN₁/dt = r₁[K₁-N₁-a_1,2N₂/K₁]

Replace 1 with a 2 for other species.

Coexistance can only occur if a is less than 1.

1. Trophic

2. Defensive

3. Dispersive