Population dynamics a good example of non-linear feedbacks in ecosystems.

r-selected (growth selected -prey) and k-selected (carrying capacity limited -predators) species:

More specifically:

K-selected species usually live near the carrying capacity of their environment. Their numbers are controlled by the availability of resources. They are a density dependent species. Food availability is the principal resource that controls population size, especially if the food is relatively localized.

The attributes of a K-selected species include a long maturation time, breeding relatively late in life, a long lifespan, producing relatively few offspring, low mortality rates of young, and extensive parental care.

The attributes of a r-selected species include a short maturation time, breeding at a young age, a short lifespan, producing many offspring quickly, small offspring, high mortality rates of young, and nonexistent parental care.

r and K selection may depend on ecosystem condition:

• Organisms that live in stable environments tend to make few, "expensive" offspring. Organisms that live in unstable environments tend to make many, "cheap" offspring.

• Extreme r-strategists, such as the oyster, lose most of the individuals very quickly, relative to the maximum life span for the species. So they must reproduce fast to survive.

• Extreme K-strategists, such as humans, most individuals live to old age (again relative to the maximum life span for the species) and thus the population continues to build until some ecosystem carrying capacity is reached (then catastrophic collapse?)

While r vs K selection is a useful framework it generally oversimplifies the problem. Most real systems are subject to non-linear or chaotic dynamics, involving unstable oscillations around points of equilibrium.

K-selected species are idealized by the equation below where it can be seen that the growth rate, R, becomes zero as N approaches K. This is called negative feedback - the idea being as conditions get "crowded" it is detrimental to growth. The problem with real systems is that estimates of K are very difficult and uncertain:

Note when N << K, N/K is zero. In this case pure (unbounded) exponential growth is occuring:

r-growth is standard exponential growth which leads to population crashes. The functional difference between r and K selected growth is shown below:

In reality, K-selected growth qualitatively looks like this:

In this case we have r-selected growth up to the carrying capacity line but then we have oscillations about that line. These oscillations have points of unstable equilibria (i.e. the smaller scales peaks and valleys). Small perturbations in the system, when species growth is at one of these points, could trigger catastrophic continued r-growth or rapid decay. This is called non-linear dynamics (sometimes called chaotic dynamics). In the case where too much habitat has been develolped (h has fallen below h_critical then the population will generally crash, as shown below:

In non-linear dynamics, small perturbations in some system can cause large changes in overall system evolution.

Some of this behavior is seen in the classic predator-prey case study of Lynxs and Snowshow hares. Its an excellent example of density dependent lag time in some system:

Note that this system is in quasi-equilibrium when averaged over long timescales. This is called Statistical Equilibrium .

On short time scales, however, the relative species densities are rapidly changing. Each peak or valley in this cyclical behavior is a point of unstable equilibrium and a small pertubation could drive the system completely out of control.

An additional complication is time lags or system response. In real life, this is the limiting factor for accurate modeling.

Non-linear dynamics is very difficult to accurately model because of the degree of non-linear response of the system is unknown and the relevant stress parameters are poorly understood.

But it can be effectively simulated.

Sometimes predator stablization can occur, if the predator can find prey over large areas. Here is a good example from Tawny Owls in the UK

How to do simple modeling: (and the previous simulation is based excatly on this sample model)

The simple model is that predator-prey interactios affect predator density and prey densities and then there is feedback between the two populations based on their densities.

Prey Population growth rate:

• If a prey population is left with no predators, it will (at least until it approaches carrying capacity) increase exponentially. This is seen many times when a species is released into a new habitat where predators are absent. So start with the exponential growth rate.
  dN/dt = rN N = N_oe^rt ; r is the growth rate (percent growth per year)

• Introduce a predator that will slow the growth rate.
  dN/dt = rN - (ones that get eaten)

• What factors determine how many are eaten?
  • How frequently the predator and prey encouter each other which is a function of the population size of predators (P) and the population size of prey (H) so this is how predator and prey density enter into this.

  • Predator efficiency at capture, p (this is similar to what the α parameter is from out previous discussion)

  • This now leads to the modified equation
    dH/dt = rH -pHP

Predator population growth rate:

• When no prey around, we can describe the decline in the predator population as mP, where the population declines exponentially at a mortality rate of m

• When prey are present, the population of predators will change like:
  dP/dt = number born - mP

• Factors relating to predation that affect the number of predators born.
  • The population of predators already present, P.
  • The population of prey, H.
  • Predator efficiency at capture of prey, p.
  • How many prey items it takes to make a new predator, or how nutritious the prey are like p, this is an efficiency (called a conversion efficiency or a below) and is a number between 0 and 1.

  • note that the product a*p is the reproduction rate of predators per 1 prey eaten. If a = 0.5 and p = 0.5 then a*p = 0.25 meaning it would take 4 eaten prey to produce a new predator. Real world values for a and p are generally not well known.

• So the predator population is expected to change as:
  dP/dt = apHP-mP

This predation model is called the Lotka Volterra model (in case you want to Google on it).

The model has two equations, one for prey populations and one for predator populations. To understand dynamics, find points when populations are in equilibrium (population growth rates of zero).

• For prey, dH/dt = rH -pHP; dH/dt = 0 occurs when rH=pHP or P=r/p

  Thus r/p is a controlling parameter

  

  For predators, dP/dt=0 occurs when apHP=mp or H=m/ap

  Thus m/ap is a controlling parameter

  

When these lines of equilibrium are combined. We see that the model predicts cycling between prey and predators.

Starting from lower left quandrant and moving counter clockwise.

1. H < m/ap P falls and H grows
2. H grows until H > m/ap then P starts growing
3. When P > r/P H starts to decrease
4. As H starts to decrease, P will start to decrease as well, until H can build again

Nominal Vectors in Predator Pray relations.

Real Blackboard lecture on this (lots of math making this look harder than it really is).

Note, what we call parameters a,m and p maybe denoted by other symbols in these other presentations of this material.