Population Dynamics

Population dynamics are essential for studying evolutionary dynamics

Polymorphism

Most populations are polymorphic: They contain phenotypic variations

Simple Scenario

We are starting from a very simple scenario, based on the following assumptions

Assumtion 1: Population definition

(Popultions are composed of conspecific individuals at a certain point in time.)

Assumption 2: Asexual Reproduction

For the core understanding of population dynamics, let's just consider asexual reproduction to reduce complexity (for now)

Assumption 3: Homogenous populations

There is no physiological variation in between individuals - but it does vary between ages (for now)

Fitness

Fitness is the expected number of direct descendants of one individual over one demographic time step (from a parent to a descendent)
- Note that descendant includes the parent

Formula:

ω = s + f_{e}

Fitness is survivorship plus effective fecundity (number of successful offspring produced over one demographic time step). This means they survive enough to reproduce.

Dynamical System

Population Dynamics are a fundamentally dynamical system - they change over time. We can model this as $n_{t + 1} = g (t)$ , where g(t) is a growth function at a given time.
General equation for a simple system:

n_{t + 1} = w_{t} n_{t}

Where $w_{t}$ is per capita growth rate (births - deaths). We

Density-independence:

If we have constant growth, then $w_{t}$ is constant (just $w$ ). The formula then is $n_{t} = w^{t} n_{0}$ .

Malthus predicted that all organisms have the capacity to grow exponentially, but since resources can only increase linearly, this inevitably results in population bottlenecks.
- This was not entirely true, since technological improvements have increased the amount of food available - and in fact population growth can also drive technological processes.

Density-dependent growth

Two fundamental forms of density dependent growth: Material resources (food) and space (habitat, light).

Competition for resources: Beverton-Holt model

One of the simplest forms of density-dependence:

w (n) = \frac{f}{1 + γ n}

Where $f \geq 0$ is the maximal fecundity (physiologically constrained) an individual can have if there is no limit to resources

And $γ \geq 0$ is a coefficient representing density dependent competition. $γ$ is zero, then there is no effect of density dependece (limiting resources).

When fitness $w$ is 1, that means the population is barely stable, after that it can no longer replace itself. It should stabilize around 1.

This model was based on the Holling Type II functional Response, but simplified so that $γ$ is driven by the Holling Type II parameters for handling time and energy conversion.

Equilibria and Stability

Similar to how we learned in maths, there are stable and unstable equilibria

$Δ n = w (n) n - n$

Plugging in the Beverton-Holt model into this:

Δ n = (\frac{f}{1 + γ n} - 1) n

!Lecture_1, p.30

As seen on the graph, the equilibria appear when the graph goes through the x-axis. There are stable equilibria where there is no change in the population (there is also an unstable equilibrium at n=0 where the population is extinct).

To find the stable equilibrium when growth $w (n^{*}) = 1$ :

w (n^{*}) = \frac{f}{1 + γ n^{*}} = 1

Rearrange to find the stable population size (or carrying capacity):

n^{*} = \frac{f - 1}{γ}

Competition for space:

Consider that there are $n_{m a x}$ breeding sites:
Each of the n adults produces on average $f$ offspring which is assumed to be smaller than the total number of breeding sites $f << n_{m a x}$
Probability of survival to the next breeding season is $s$ , so a total of $n s$ adults survive

Individual fitness

w (n) = s + (1 - \frac{s n}{n_{m a x}}) f

The non-trivial equilibrium for this model is:

n^{*} = n_{m a x} (\frac{1 - s - f}{f s})

This is called "lottery competition" where everyone is competing for a limited number of spots.