Lecture 3 – Population Modelling

Mathematical Models

Also known as analytical or theoritical, are designed for general insights. This includes making predictions, generalisations, and so on. It has to be reproduceable (to be tested and validified) as well as frame questions to be addressed

Uses of Mathematical Models

Allows us to:

• Expose/make us think about and test our current assumptions and notions

• Provide testable consequences and or qualitative/quantitative new insights

• Easily transfer knowledge

Deterministic Modelling

Refers to the controlled sort of modelling where everything is set up (parameters, equations, functions, and so on) so that a single input will always result in the same single output

Randomness is not considered, thus only resulting with one outcome

There are some benefits to using deterministic modelling:

• Is easier and takes less resources to be reproduced. The lack of randomness gives it more predictability

• Can help establish clear cause-and-effect relationships

• Easier to understand and share information with

Exponential Growth Model (Deterministic)

N_t+1 = N_t + B – D + I – E 

B = Births, D = Deaths, I = Immigration, E = Emigration, t = Time

The equation above essentially states that the population of the next time step will be the current time step’s population plus births & immigrations and minus deaths & emigrations.

When immigrations and emigrations are the same, we can say that ΔN = B - D.

Exponential Growth Model: Continuous Differential Equation

We need to put this rate over a time interval to give us the continuous differential equation to model the change that happens at every instant.

dN/dt = B - D where B = bN, D = dN 

• b = instantaneous birth rate (births/(individual * time)) 

• d = instantaneous death rate (death/(individual * time)) 

This is then further dimplfied into

dN/dt = (b - d)N or rN

• r = Instantaneous growth

r > 0 (growth)

r < 0 (decline)

r = 0 (no growth)

It is then further simplfiied and transformed into the continuous growth

N_t = N₀exp^rt

Exponential Growth Model: Discrete Differential Equation

Or, it could be transformed into a discrete differential equation that allow us to figure out the change at specific time intervals and figure out the population after a set amount of time

N_t+1 = N_t + r_dN_t 

Could also be modified to give us doubling times, how much time it takes for a population to double

Exponential Growth Model: Assumptions & Limitations

There assumptions that allow the model to work:

• No immigration or emigration (closed population)

• Constant birth & death rates (with unlimited resources, no randomness)

• No age, size, or genetic structure/differences (individuals are all the same)

These assumptions are limitations in itself as in real-life environmental systems, there tends to be randomness, carrying capacities, differences amongst individuals, and so on

• The main limitation is that populations don’t grow/change forever

Logistic Growth Model (Mainly Deterministic)

Shows the reality where populations do have limited resources and reaches its carrying capacity K.

dN/dt = rN x (1 - N/K)

• rN = growth or N_t + r_dN_t = discrete growth

• (1 - N/K) = reduction due to density

There are many examples in real-life that ressemble the logistic growth model (bacteria populations, population of wild animals, and so on)

• Depending on the system being looked at (simple or complex), the way the population grows may ressemble the logistic model more or it could fluctuate around K (due to random chance)

Change will be more intense before carrying capacity is reached. Then, it’ll start to slow down as it approaches K where it’ll be essentially zero change.

Logistic Growth Model: Assumptions

Also has some assumptions

• A constant carrying capacity (where resource limits don’t change)

• Linear density-dependence (individuals have same effect on per capita growth)

Stochastic Modelling

Refers to models that do take random chance into account which then results in a variety of possibilities, each having a probability of occuring.

Applying this to population growth, we’d expect random fluctuations in the growth rate of the population alongside taking carrying capacity into account

N_t+1 = N_t + r_d(𝜉(t))N_t 

• 𝜉(t) = random number changing every time step (t) W

When simulated multiple times, we’d see that the way a population grows wouldn’t proceed the same between the simulations. Rather than it being a deterministic smooth curve, there would eb random jaggedness to it

With these models, we can make histograms of different possibile outcomes and the probabilities associated with each outcome.