Differential equations

Also known as: continuous time models, stock and flow models (not always), population models (not all), continuum models, ordinary differential equations (ODEs)

Variants: partial differential equations, delay differential equations, stochastic differential equations

One sentence description: Represent the rate of change of a continuous system state at an instantaneous moment in time. Can be solved to show the continuous change of the state of the system over time.

Key references:

Key examples:
Infectious disease spread in host populations (e.g. susceptible-infected-recovered; SIR models), classical models of animal population dynamics (e.g. the logistic equation), predator-prey models (e.g. Lotka-Volterra)

Description: Classically these represent how various biological and environmental states influence the rate of change of a continuous state at a given moment in time. They typically have the mathematical form

[Rate of change of property at time, t] = [rate of inputs] – [rate of outputs]

For example a simple population model might have the form

[Rate of change of the density of individuals in an area at time, t] = [birth rate] – [death rate] + [immigration rate] – [emigration rate]

For example, the rate of change of population size is typically predicted using expressions for the rate of births and deaths.

Ecology typically predicted: changes in the quantity of ecosystem states that can be said to change continuously over time (e.g. concentrations of individuals per unit area, mass). These can be extended to represent spatiotemporal dynamics (often using partial differential equations) and age/size structured dynamics (e.g. integro differential and matrix equations).

Pre-requisite skills: These usually require some background mathematical training in order to understand, formulate and simulate. An understanding of differential equation solvers, basic algebra, the ability to read mathematical expressions helps in using differential equation models.

Differential equations are relatively straightforward to define using mathematical expressions (they are easy to write down). They are particularly convenient for concisely specifying how the rates of ecological processes depend on various biotic and abiotic factors. There is also a substantial body of mathematical theory to help analyse the properties of differential equation models, such as the existence of steady states and their stability. There are many computational libraries and software tools to aid in constructing, analysing and simulating differential equation models (including libraries in R).

Limitations: Two key limitations are (1) the appropriateness of the continuum assumption for the system being modelled in terms of the modelled property varying on a continuous (rather than discrete) scale and (2) the need for appropriate simulation algorithms to obtain the correct solution to the differential equation. In relation to (1) users should watch out for the modelled quantities reaching unrealistic values (e.g. 0.0001 individuals per square metre might translate to less than one individual across the modelled area which might correspond to extinction in reality). In relation to (2) checks should be made that the simulation method does not lead to numerical artefacts (e.g. it is unwise to simply sole the differential equation as a difference equation without understanding the effects of the chosen step size). It is also hard to appropriately incorporate representations of stochastic processes in differential equations (see stochastic differential equations).

Data requirements: Often the parameters used to represent biological rates in differential equation models are obtained from empirical studies. It can be technically difficult to quantify the instantaneous rate of change of a process and various technical methods have been tempted to do this. More recently studies have begun to infer parameters using various model-data fusion techniques (e.g. Bayesian parameter estimation). Differential equation models almost always represent the average dynamics of the system being modelled so one should expect noise about the predicted line when model predictions are compared to empirical data.

Resources: a number of software packages have built in or downloadable methods for simulating and analysing differential equation models. The most popular examples are MATLAB, Mathematical and R.

Validation: Checks should be made that the assumption of continuous variation in the modelled property is valid (check size of low numbers) and that unwanted numerical artefacts are avoided.

Other Uses: Differential equation models have a long history in other areas of science, particularly physics, chemistry, maths and engineering. They are often used to represent changes in the concentrations of substances.

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