Ch.4 Flashcards
Biological organisms are set up to
reproduce.
Our species has experienced dramatic growth
Population Growth Models ( 2 types)
Unlimited resources (exponential growth model)
Limited resources (logistic growth model)
Compensatory dynamics
the idea that in a resource-limited community, an increase in the density of same species should be balanced by a decrease in the density of other, interacting species-is the basis for most theories of community stability and ecosystem resilience.
Exponential Population Growth: unlimited resources
Nt = number of individuals in a population at time t
Nt+1 = the size of the population at the end of the next time interval
B = number of individuals born
D = number of individuals that die
I = number of individuals that immigrate into the population
E = number of individuals that emigrate from the population
At time t +1 the population size can be estimated by:
Nt+1 = Nt + B + I + - D - E
see slide for more
Exponential Population Growth: unlimited resources
to determine the difference between one time point to the next one must:
Nt+1 - Nt = B + I - D - E
unlimited resources
to determine the difference between one time point to the next ( closed population) E = 0 & I = 0
Nt+1 - Nt = B - D
B and D will be dependent on the
number of individuals in the population.
population size vs time - J curve
all you need to recognize is that the change in the number of individuals with regard to time is: dN/dt
dN/dt = B - D
What does b and d stand for
we will use b as the per capita birth rate and d as the per capital death rate. And this will yield:
dN/dt = bN – dN
Or
dN/dt = (b-d)N
r =
Instantaneous rate of increase, the population growth
is governed by this constant
the population growth
is governed by which constant
r = Instantaneous rate of increase
When resources are unlimited, r, is at
its maximum value
for a species and this is referred to the species’
intrinsic rate of increase.
max = 1.0
intrinsic rate of increase
the number of births minus the number of deaths per generation time—in other words, the reproduction rate less the death rate
K is representative of
the carrying capacity
the number of individuals the environment can support indefinitely, given fluctuations in resources in the environment
carrying capacity
the number of individuals the environment can support indefinitely, given fluctuations in resources in the environment
Logistic Population Growth
as a population grows, and its density increases, a number of things can happen: DACA
Depletion of resources
Accumulation of waste products
Aggression between individuals
Competition for mating opportunities
Density Dependence
Depletion of resources
Accumulation of waste products
Aggression between individuals
Competition for mating opportunities
All of these negative feedback inputs between population size and per capita growth rate has been termed density dependence.
b’ and d’ now represent
the per capita density-dependent birth rate and per capita density-dependent death rate, respectively.
logistic equation
describe the curve
dN/dt = rN(1-N/K)
S curve the levels off at K
carrying capacity graph
Increased birth and decreased death
decreased birth and increased death
^ shift back to equilibrium of K
exponential curve looks like a
J curve
How does the population growth rate vary as a function of population size?
when the population is at 1/2 its (K) carrying capacity it has the most growth ( dN/dt)
What about the per capita population growth rate with respect to population size (N)?
Negative Slope
linear decreease
Assumptions of the Logistic Growth Model
( rarely met in nature)
- The per capita growth rate is a linear function of N.
- The population growth rate responds instantly to changes in N. (no time lags)
- The external environment has no influence on the rate of population growth.
- All individuals in the population are equal (No effects of individual age or size).
Logistic Growth Model is difficult to meet but is still taught bc
It simply shows that population density can alter the growth rate of populations.
Also because it can be modified to relax the assumptions
Woiwod & Hanski (1992)
studied density dependence
Found density dependence in 69% of aphid and 29% of moth time series data sets.
BUT, when short time series (<20 years) are removed from the analysis, the percentage of studies showing significant density dependence increased: 84% of aphid, 57% of moth
Turchin and Taylor (2006)
Found density dependence in 18 of 19 forest insect populations and 19 of 22 vertebrate populations
Also observed cyclic, chaotic, and stable dynamics
Brook and Bradshaw (2006)
Found “convincing evidence for pervasive density dependence in the time series data” and “little discernible difference across major taxonomic groups”
They also found a trend toward zero net growth (implying equilibrium) as the number of generations in the time series increased.
Brook and Bradshaw,
2006
graphs
In (A) density dependence increases with the length of time series data collected
In (B) as the number of generations monitored increases net rate of population change tends toward zero (equilibrium)
the evidence for density dependence is
consistent and overwhelming for many different types of species.
Density dependence is important for
population regulation
Allee Effects
at low population densities animals have issues with things like finding a mate
allee effects are a positive association between fitness and population size.
Two manifestations of allee effects
- Component Allee Effects – positive association between a fitness component (i.e. survivorship, fecundity) and population size
- Demographic Allee Effects – when component allee effects cause a positive association between population size and per capita growth rate.
starfish example
Due to escape from predation, there is a significant amount of allee effects occurring in starfish populations.
define compensatory Dynamics
The idea that the density of all individuals within a functional group should be a conserved quantity has been termed “compensatory dynamics” or sometimes also called “community-level density dependence” or even “zero-sum dynamics”
Taking a look at densities at the community level, predicts that as one species abundance decreases other species that have a high degree of niche overlap should increase.
How does the abundance of one species within a function group (i.e. trophic level) affect the abundance of other species within that same group?
This question looks at
compensatory Dynamics
( the impact of each species on each other in terms of abundance)
James Brown
Focuses on comparing natural dynamics of the rodent communities in the control plots with those in the kangaroo rat removal plots.
Found that:
1. there was a shift from grassland to shrubland species
2. Species richness remained quite consistent, despite significant species turnover
3. Decrease in rodent body size, but an increase in rodent abudance
concluded that compensatory dynamics were occurring in their plots.
Are common or rare species more likely to experience density dependence?
-common species showed little variation in abundance and significant density dependence.
-Rare species were short-lived in the ecosystem and showed low density dependence
observational graph from book
the steeper the line the higher the density dependence
The Lotka-Volterra Model
assumes that the prey consumption rate by a predator is directly proportional to the prey abundance. This means that predator feeding is limited only by the amount of prey in the environment.
simple assumption of Lotka-V
One assumption of this simple model is that prey populations grow exponentially.
( J curve mdoel)
equation for exponential growth
dN/dt = rN
Where N = number of individuals in the prey population and r = prey’s per capita population growth
rate
If predation is the only source of mortality for a prey population and the predator has a
linear (type I)
functional response (which means the number of prey killed is directly proportional to prey density),
then we can alter the equation to be this:
dN/dt = rN – aNP
Here a = predator’s per capita attack rate and P is the number of predators
IN THE ABSENCE OF PREY A predator population would be expected to
decline exponentially
at the rate of:
dP/dt = -qP
Here, q is the predator’s mortality rate (which is density-independent) and is measured as the number of
deaths per predator per unit time.
growth rate of predator population
which depends on the number of prey consumed
per unit time, is equal to: aNP.
f =
eating prey will turn into new predators at a constant rate of f.
eating prey will turn into new predators at a constant rate of f. So, birth of the
predator population is equal to:
faNP
r =
b prime - d prime
prime = per capita
dynamics of the predator population is described by:
dP/dt = faNP-qP
Net change in predator populations
dP/dt = faNP-qP
Net change in prey populations
dN/dt = rN – aNP
two equations that make up Lotka-Volterra Predator-Prey Model
dN/dt = rN – aNP
Net change in prey populations
And
dP/dt = faNP-qP
Net change in predator populations
ZERO growth isocline aka
equilibirum
ZERO growth isocline
when a population is neither growing nor
declining.
dN/dt = 0
Prey Population Dynamics
Reach zero growth isocline when dN/dt = 0 and the prey population will be
at equilibrium when the density of predators P = r/a (remember that r =
prey per capita growth rate and a = attack rate).
( horizontal lines0
prey population will be at equilibrium when density of predators is equal to
P = r/a (remember that r =
prey per capita growth rate and a = attack rate).
Predator Population Dynamics
Reach zero growth isocline when dP/dt = 0 and the predator population
will be at equilibrium when the density of prey N = q/fa (remember that q =
number of predator deaths, f = rate of conversion of consumed energy into
new predators).
( vertical lines)
PUTTING PREDATOR AND PREY ISOCLINES TOGETHER…
It is the number of predators present that determines whether the
prey population is at equilibrium AND it is the number of prey
present that determines whether the predator population is at
equilibrium.
Placing the two graphs on top of one another yields four regions of
predator-prey state space, the narrow arrows show the response of
the predator and prey populations in each region. Heavy diagonal
arrows show the combined trajectories of the predator/prey
populations in each region. Over time, the predator and prey
populations will cycle continuously (seen in the ellipse).
prey-predator dynamics looking only at density
cyclical action - see slide image
We can use some of the information gleaned in the mathematical
models to determine how much one can harvest a population without
making it go extinct.
when to harvest
1/2K ?? ask
( see graph)
harvest above equilib. will generally cause pop to go extinct and instability
tragedy of the commons.
This idea describes a situation where each entity
harvesting an open resource receives the direct benefit of harvesting that resource, whereas all
consumers of the resource share the cost.
