Unformatted text preview: Density Dependent versus
Density Independent Growth According to the first law of population ecology, populations growing without restraints, will grow “geometrically” or “exponentially” depending on the life history. Density Independent Growth This is known as density independent growth
Growth is determined by a fixed parameter
R, λ or r Density Dependent versus
Density Independent Growth Population growth may be curtailed by “density independent factors”
Fires, storms, floods, landslides etc.
Mortality occurs irrespective of the present size of the population. Density Dependent versus
Density Independent Growth Density dependent growth, however, assumes there is a limiting factor in the environment (food, nutrients, water, nest sites etc.) such that the growth parameter is increasingly modified as the population approaches a “carrying capacity.”
Hence: density dependent growth. Density Dependent Growth The exponential growth models assumed that growth is Density Independent. BUT…
most Biological Populations do not show Exponential Growth for long.
For example, a Paramecium Population. Density Dependent Growth Based on data such as these, we conclude that populations do not have unlimited resources.
Population growth will cease Liebig’s Law of the Minimum (1840, 1855) “Under steady state conditions, the population size of a species is constrained by whatever resource is in shortest supply.” Density Dependent Growth Logistic Model:
Population grows until it reaches the carrying capacity (K) of the environment for the population. Density Dependent Growth Carrying capacity is defined as the number of individuals of a population that can be maintained indefinitely in a specific environment. Logistic Equation
Logistic Equation The logistic equation is based on the law of self limitation, where K = a carrying capacity. K − N dN = rN
K
dt Logistic Equation
Logistic Equation Assumptions:
1. 2.
3.
4. Birth and death rates vary linearly with population density.
K is a constant.
There are no effects of age distribution.
Immediate interaction between carrying capacity and population size. Logistic Equation
Logistic Equation If we add a “lag” time the population can exhibit a number of behaviors. Actually the behavior is determined by the interaction (product) of the intrinsic rate of increase, r, and the lag time, T (tau).
The larger the product the more unruly the population. Effects of Lag Times
Effects of Lag Times A small product or a Tau of zero produces the usual form of the logistic with a stable point at K.
A larger product produces a small oscillation before settling in at K.
A still larger product produces a series of oscillations that dampen to K. Effects of Lag Times
Effects of Lag Times An even larger product produces a permanent oscillation around the carrying capacity known as a stable cycle.
A very large product produces ever increasing oscillations resulting in population extinction.
On the next slide r = 0.3 is combined with various lag times. 4000
Tau = 0
Tau = 3
Tau = 4
Tau = 6
Tau = 9 Population, N 3000 2000 1000 0 0 25 50
Time 75 100 Effects of Lag Times
Effects of Lag Times If we hold Tau constant at 3, we can look at the effects of different rvalues on the behavior of the population.
First, a simple logistic curve. Tau = 3 r = 0.1
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Time 80 100 120 Effects of Lag Times
Effects of Lag Times Next dampened oscillations to a stable point. A stable or limit cycle. And finally extinction. Tau = 3 r = 0.3
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Time 80 100 120 Tau = 3 r = 0.5
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Time 80 100 120 Tau = 3 r = 1.0
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5000 Population Siz e 4000
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Time 80 100 120 Chaos and the Behavior of the Chaos and the Behavior of the Discrete Logistic Model Below is the Ricker equation for populations undergoing discrete population growth. Time lags are implicit in the discrete logistic model. N t +1 = N t e K − Nt r
K = Nt e Nt r 1− K Chaos and Behavior of the Discrete Model In actuality, if the growth parameter is large enough, populations will behave in unusual and unexpected ways even without time lags.
With large rvalues we can have two
point, fourpoint, and eightpoint cycles. And once r > 2.692 (a very large value!) we have what is known as chaos. Behavior of the discrete logistic model: stable Behavior of the discrete logistic model: stable equilibrium point when r < 2.0
600 Population, N 500
400 r = 0.50
r = 1.5 300
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Time 50 60 70 Behavior of the discrete logistic model: two Behavior of the discrete logistic model: two point cycle when r = 2.20 Population, N 800 600 400 200 0
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Time 50 60 70 Four point cycle when r = 2.60.
Four point cycle when r = 2.60. Population, N 1200 900 600 300 0
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Time 50 60 70 Behavior of the discrete logistic model: Behavior of the discrete logistic model: chaos when r = 2.75.
1200 Population, N 1000
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Time 50 60 70 Summary
1. 2. Models for density dependent population growth can be derived starting with both difference and differential equations.
All models assume that the growth
rate parameter is dampened as the population approaches a carrying capacity. Summary
1. 2. Models for density dependent population growth can be derived starting with both difference and differential equations.
All models assume that the growth
rate parameter is dampened as the population approaches a carrying capacity. Summary
3. 4. 5. Modifications can include the inclusion of time lags.
Populations with time lags show a variety of behaviors found in nature (limit cycles, boom and bust cycles etc.)
We should not expect populations in nature to remain constant from one year to the next. Summary
6. The logistic equation should be thought of as a starting point for introducing the idea of limits to population growth. Questions? ...
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 Summer '11
 Crerar
 Harshad number, Tau, discrete logistic model

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