MBW:Density Dependence in Single-Species Populations

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This article is a review of the paper Density Dependence in Single-Species Populations by M.P Hassell published in Journal of Animal Ecology, Vol. 44 No. 1, 283-295 (1975).

by Eric Greenwald and Scott Tatum

1 Density Dependence in Single-Species Populations

Density Dependence in Single-Species Populations

Abstract

There have been many attempts to describe density dependent populations throughout history. Most models can only describe portions of the density dependence, such as the high density portion of the population, but have failed to be able to fully describe any data. The objective of this new model it to combine features of previous models to get a "complete" model of density dependence in populations. This article discusses the old models and the drawbacks of each of these, then discusses the new model and its stability, and finally discusses the application and results of the new model.

Context

Density Dependent Relationships

There have been several models that describe how single-species populations grow based on their density. In general populations exhibit very different dynamics for high density and low density populations. This relationship can be seen in the Density Dependent Relationships to the right. Each figure is a plot of $log[\frac{N_t}{N_s}]$ vs $log[N_t]$ where $N_t$ is the initial population density and $N_s$ is the surviving population density. Figure 1 below shows two cases from the Density Dependent Relationships figure.

Figure 1: Two examples of density dependent populations.^[1]

As one can see the populations growth is greatly effected by the starting densities. There is a critical density $N_c$ where the k-values change slope drastically from low to high.

History

Density Dependent Model

The above figures show several density dependent relationships that arise from intraspecific competition for a fixed amount of food. Varley & Gradwell ^[2] and Haldane^[3] express the mortality as $log(N_t/N_s)$ (also known as the k-value) which is a ratio of the starting density to the surviving density.

A simple model to describe a density dependent relationship is given by Varley & Gradwell^[4] and Morris^[5] is

$N_s = \frac{1}{\alpha}N_t^{(1-b)} \quad \quad \quad \quad (1)$

where $\alpha$ and b are constants that relate the mortality with the population density. Putting this into logarithmic form and rearranging we can get

$log \left(\frac{N_t}{N_s}\right) = log(\alpha) + b\,log(N_t) \quad \quad \quad \quad (2)$

This model is only good for the linear portions of the data and cannot model the transition between low and high density relationships seen in Figure 1. May^[6] examined Equation 1 using a net rate of increase $\lambda$ to give the equation

$N_{t+1} = \frac{\lambda}{\alpha} N_t^{(1-b)} \quad\quad\quad\quad(3)$

where $N_{t+1}$ and $N_t$ are population sizes at successive generations. May^[6] also examined the stability and showed that it is governed solely by the parameter b. The population is stable when $2> b> 0$ . When $1> b> 0$ the population decays exponentially to equilibrium while $2> b> 1$ will "overcompensate" and exhibit damped oscillations back to equilibrium. If $b> 2$ the model predicts increasing oscillations.

The model in Equation 3 has a limitation of being linear on on log scale. This creates a problem when the line intercepts the x axis. This would give a threshold population density, $N_c$ (seen in the Density Dependent Relationships figure f). This is only possible when $log(N_t/N_s)$ is negative meaning that there is a larger surviving population than the initial population, which would require either reproduction or immigration prior to the sampling of the survivors. A corrected model was developed by Varley, Gradwell & Hassell^[7] with a simple correction

$N_{t+1} = \frac{\lambda}{\alpha} N_t^{(1-b)}$	$\mbox{for} \quad N_t>N_c$	$(4)$
$N_{t+1} = \lambda\, N_t$	$\mbox{for} \quad N_t<N_c$	$(4)$

This model solves the problem of $N_s > N_t \mbox{when} N_t < N_c$ but if $b > 2$ the population will oscillate irregularly around the equilibrium with a pattern dependent on the initial condition which is not something that would occur in nature. The expected outcome is one of stable limit cycles around the equilibrium point which can be achieved when "smoothing" is introduced. May^[6] created the model

$N_{t+1} = (\lambda \, N_t^{-f(N_t)})N_t \quad\quad\quad\quad(5)$

where $f(N_t)$ replaces the density dependence term b. The function is defined with the limiting forms

$f(N_t)\rightarrow 0$	$\mbox{when} \quad N_t\ll N_c$	$(6)$
$f(N_t)\rightarrow b$	$\mbox{when} \quad N_t\gg N_c$	$(6)$

Scramble and Contest

Intraspecific competition can be defined in two ways: scramble and contest^[8]. Scramble is when there is a equal partitioning of resources and there is an abrupt change from from complete survival to 100% mortality. This can be defined as

$b=0$	$\mbox{when}\quad N_t<N_c$	$(7)$
$b=\infty$	$\mbox{when}\quad N_t>N_c$	$(7)$

In contest each successful animal get all the resources it requires and unsuccessful animals do not get sufficient resources for survival^[7]. This is usually demonstrated by a fixed number of refugees where the surviving number remains constant while the population increases. This leads to a density dependence given by

$b=0$	$\mbox{when}\quad N_t<N_c$	$(8)$
$b=1$	$\mbox{when}\quad N_t>N_c$	$(8)$

The figure below shows an example of each case just described.

Figure 2: Density Dependent Relationships for Extremes: a) Scramble, b) Contest.

This model aproaches the logistic equation except for the discontinuity at $N_c$ . Since these two examples are the extremes the normal situation will lay somewhere in the middle with $1>b>0$ when $N_t > N_c$ would represent different amounts of competition and $\infty>b>1$ when <N_t > N_c </math> represents a combination of scramble and contest. Figure 1 shows this range between the multiple figures.

A model that describes the extremes of scramble has been presented by May^[6] as a limiting case of a population of periodical cicadas has the form

$N_{t+1}=\lambda \, N_t e^{-aN_t} \quad\quad\quad\quad(9)$

The mortality rate increases exponentially as $log(N_t)$ increases and has the limiting forms given by Equation 5 of

$f(N_t)\rightarrow 0$	$\mbox{when} \quad N_t\ll N_c$	$(10)$
$f(N_t)\rightarrow \infty$	$\mbox{when} \quad N_t\gg N_c$	$(10)$

The stability properties were shown by May^[6] to be governed by $\lambda$ alone. As the value of $\lambda$ increases the population exhibits exponential damping, damped oscillations, and stable limit cycles respectively. The drawback of this model is that it does not fit models where density dependence is linear at high population densities. The goal is to combine these above models to get one that will fit linearly at high densities but model the transition around $N_c$ properly.

Mathematical Model

Hassell proposes a new density dependent model that smooths the transition between across the critical population boundary. This new model, of the following form $N_{t+1}=[f(N_t)]N_t$ , has to have the following constraints.

$f(N_t)\rightarrow 0 \quad \mbox{when} \quad N_t \ll N_c$	$(11)$
$f(N_t)\rightarrow b \quad \mbox{when} \quad N_t \gg N_c$	$(11)$

To be able to satisfy these conditions as well as constraint that $N_t/N_s\rightarrow 0$ when $N_t\rightarrow 0$ a logistic function was used by Hassell with the following form.

$N_{t+1}=[\lambda (1 + a N_t)^{-b}]N_t \quad\quad\quad\quad (12)$

Where $\lambda$ and $b$ , the same same as above, are variables representing the net rate of increase and the high density slope, respectively. The new parameter, $a$ defines the threshold density so $a=1/N_c$ .

Example of New Model

The Importance of this model change can be seen in the "Example of New Model" figure (right) that the extremes follow the defined limit requirements but the intermediate region is a smooth transitional mix between the two.

Stability Analysis

First, the equilibrium point is defined where $f(N^*)=1$ . The stability of this system is determined by looking at the slope of $log(N_t/N_{t+1})$ with respect to $N_t$ at the equilibrium point $N^*$ . May et al. showed that when $0<\tilde{b}<1$ the system experiences exponential damping, $1<\tilde{b}<2$ the system experiences damped oscillations and when $2<\tilde{b}$ the system has stable limit cycles , where ^[6]

$\tilde{b}=-N^* \left[\frac{df}{dN}\right]^*=b((1-\lambda^{-1/b})\quad \quad\quad\quad (13)$

The above equation can be plugged into the stability limit to be able to derive a stability relationship between $b$ and $\lambda$ . These relationships are shown in the figure below.

Figure 3: Stability dependence on model parameters ^[1]

The figure below shows for a fixed a and b when you vary $\lambda$ you can see the different types of stability.

Figure 4: Plots of N against generation where a=0.01 and b=4 and varying $\lambda$ ^[1]

Model Application

Application of Model to Data

Hassell then applied his model to some data for different insect populations and showed that the model worked for multiple cases even though some did not fit very well. Hassell showcases some of these models in the figure to the right (Application of Model to Data").

The main issue with much of the experimental data is that the range of population densities that are often studied are not broad enough so the data may not cross this critical density boundary. None the less the model can still be used to fit these data fairly well.

Discussion

The primary bonus of this new model is that it has a smooth transition across the critical density. One major weakness of this model is that the abrupt transition that has been observed in some insect types as can be seen in Figure 1 a. This weakness is considered minor because "under field conditions, such sharp discontinuities as seen in Fig. 1 are much less likely" ^[1] (see "Density Dependent Relationships" for reference to "Fig 1")

The stability analysis also leads to the conclusion that the model result significantly depends on how you approximate the variable $\lambda$ because as shown in Figure 3, as long as $b>1$ the stability regime will depend on $\lambda$ . Therefore Hassell says that you cant just assume $\lambda$ is equal to the fecundity, instead one needs to take into account all of the mortalities during the life cycle. One example proposed by Hassell is when looking at $\lambda$ for pupae.

$log \, \lambda = log \, Fec - \left[log\, \frac{Eggs(t)}{Larvae(t)}\right]-\left[log\, \frac{Larvae(t)}{Pupae(t)}\right] \quad (14)$

Another useful area that this model can be used is looking at how perturbations. Normally, when a system is perturbed past an equilibrium solution, there tends to be overcompensation but this model can see both the oscillation that occurs with over compensation or see smooth growth to the equilibrium. This is due to the fact that this model can have both low and high density solutions, and that the stability varies for both $b$ and $\lambda$ . Thus for low values of $\lambda$ even with a pertubation we will still be in the exponential damping region thus no oscillations will occur

References

↑ ^1.0 ^1.1 ^1.2 ^1.3 M.P Hassell, Density Dependence in Single-Species Populations, Journal of Animal Ecology, Vol. 44 No. 1, 283-295 (1975)
↑ G. C. Varley and G. R. Gradwell, Key Factors in Populations Studies. Journal of Animal Ecology, Vol. 29, 399-401. (1960)
↑ J. B. S. Haldane, Disease and Evolution, Ric. Sci. Vol. 19, 3-11. (1949)
↑ G. C. Varley and G. R. Gradwell, Recent Advances in Insect Population Dynamics. Ann. Rev. Ent. Vol. 15, 1-24 (1970)
↑ Single-Factor Analysis in Population Dynamics. Ecology, Vol. 40, 580-588. (1959)
↑ ^6.0 ^6.1 ^6.2 ^6.3 ^6.4 ^6.5 R. M. May, Ecological Systems in Randomly Fluctuating Environments. Progress in Theoretical Biology, Academic Press, New York. (1974).
↑ ^7.0 ^7.1 G. C Varley, G. R. Gradwell, M. P. Hassell. Insect Population Ecology. Blackwell Scientific Publicatio, Oxford. (1973)
↑ A. J. Nicholson, An Outline of the Dynamics of Animal Populations. Australian Journal of Zoology, Vol. 2, 9-65. (1954)

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