Mean+Field

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=Mean Field Approximation / Assumption=

One [|analytical] drawback of CA models is their unpredictability and therefore the need to **let them live their lives** (i.e. run the simulation and observe the outcome). This can be very **time consuming**. Moreover, each simulation is a specific case (parameter condition) because it is necessary to fully specify the model. In order to do **parameter sweeps** one must therefore do many simulations. All in all this can be quite cumbersome which has led to attempts to obtain [|mean field approximations] for CAs in order to make a short cut (i.e. approximate CA behaviour with an ODE).

Mean field approximations are done by short cutting on localness and therefore assuming **well-mixed** and **continuous variables**. Implicitly one therefore assumes that the **local pattern** doesn't matter and that **stochasticity** and **discreteness** are also irrelevant.

As an //example//, we construct the MFA of a simple birth-death CA: Consider a binary CA where the state of a gridpoint represents the absence/presence of an individual. If a gridpoint is empty, it draws a random neighbour, and if this neighbour is an indivual, it reproduces into the empty square with probability ("birth rate") //b//. Furthermore, filled gridpoints "die" with probability //d.// Let N be the total population size. Birth events take place if an empty square "meets" an individual. Hence, the MFA is given by: dN/dt = //b// N E/T - //d// N, where E is the number of empty patches, and T is the total number of patches in the grid. If we scale T=1 and let N be the population density, the MFA simplifies to dN/dt = //b// N (1 - N) - //d// N = (//b-d//) N - //b// N^2, which is standard logistic growth.

One obvious question is whether the behaviour of such an approximation is similar to the original CA. The short answer is **no**. In such ODE approximations all individuals are assumed to see the same, "average" neighbourhood, while in CAs individuals vary in the neighbourhoods they are surrounded by. In order to compensate for this shortcoming MFAs have been constructed which try to incorporate some information from the local neighbourhood. These are called **pseudo-spatial models** (e.g. models by Tilman ([|1994])). Such models are sometimes erroneously referred to as spatial models, since they only incorporate the first order effects of local neighbourhoods. This means they still ignore **pattern formation** and growth at **edges** which is often crucial in determining dynamics. Even higher-order approximations (that include more of the "local" neighbourhood) do not help to alleviate this problem since they still only at best incorporate local neighbourhood effects. Using ODEs as MFAs will therefore never be a good replacement of CAs.

However, using ODEs as **mean field //assumptions//** can be very useful. On the one hand as a **well-mixed extreme** with which to contrast a **non-mixed extreme** (CA), i.e. using both formalisms as a [|paradigm] system. Moreover, in some cases ODEs may be a better description of a system (e.g. molecules in a cell) with respect to extreme localness in a CA. However, at this stage probably 90% of models are still ODEs, but this does not reflect their appropriateness.

An important point here is therefore the concept of using different model formalisms to study extreme cases and not necessarily to describe each others behaviour, which can never be done perfectly.

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