In an intensive agricultural system growing cereals, in which the field is well ploughed, the growth responses of the plants to resources well known, and all the plants effectively of the same age, we can treat all plants as the same and the population as one unit, This allows us to predict the population dynamics, or more typically, the yield, fairly accurately.
Outwith this simplified scenario, how do we obtain a predictive model of plant population dynamics? Most importantly here – how do we predict the dynamics of wild plant populations? In wild populations there can be significant differences between individuals. This can be due to heterogeneity in the factors important to plant growth as well as differences in the time individuals have been growing for, and genetic differences. Obtaining predictive models for these scenarios is more difficult because individuals in the population are experiencing different neighbourhoods and have had different growth histories. There are two key challenges here i) understanding the nature and magnitude of spatial heterogeneities through time and ii) incorporating such heterogeneities into plant population models. However, over the last few decades, progress has been made in these directions and the literature now contains many examples of spatial plant models.
“Spatial Ecology” refers to the study of the effects of spatial heterogeneities on ecological dynamics. Its study really took off after computing power was sufficient to allow simulations of spatial effects. In this tutorial you will experiment with three spatial models to understand the potential importance of spatial interactions and have obtained some experience in ways to account for spatial differences. The first two examples are toys, and are not ecological models. These simply illustrate the potential importance of the nature of localised interactions on the system dynamics. The third example is an ecological model – a spatial two plant species competition model. I recommend you spend a small bit of time playing with the cellular automata models before then moving on to the ecological model.
This is sometimes termed ‘elementary cellular automata’ and is a straightforward way to demonstrate how changing the simple rules of neighbourhood interaction can lead to huge differences in the dynamics
This follows on the same principles as in the 1 dimensional example. The default rule set of stay alive (black) if you have 2 or 3 neighbours, become dead (white) if you have less than 2 or more than 3 neighbours, and become alive (black) if you have exactly 3 neighbours is the famous “Game of Life” rule set invented by John Conway.
There are two very similar tools associated with this part of the tutorial. There is the Silverlight application which is accessible by clicking the link above and there is the WPF application which is accessible by clicking HERE. Please note however that there is a slight problem with the WPF application causing it to run too slowly on most computers (for most people's patience!). This will be fixed soon.
In their paper, Crawley and May (1987) investigated the conditions under which two ‘species’ of plants could co-exist:
1) An ‘annual’ species that lived for only one year and was propagated only by random seed dispersal
2) A longer lived ‘perennial’ species that was propagated only by clonal propagation into nearby space.
Importantly, throughout this model the perennial is completely competitively superior to the annual for space. This means that if a perennial occupies a site then the annual cannot exist in that site, and the existence of an annual in a site has no effect on the probability that a perennial will invade that site. Space in this model is represented as a lattice of sites, identical to the way space is modelled in the two dimensional cellular automata. Only one individual plant can occupy a site at any one time. Time is modelled as discrete one-year time steps.
I have selected this study because of several convenient features.
1) It is one of the first spatially explicit models in plant ecology – each individual occupies a certain position in space, which affects the probability of events occurring.
2) The authors give almost full details of their assumptions, making it easy to reconstruct their methods.
3) They also accompany the spatial model with a mathematical approximation of the dynamics which allows further insights to be gained, which emphasises the importance and limitations of analytical approximations.
4) The findings leave a lot of potential avenues for future research that are worth discussing.
5) In the subsequent decades, a lot of subsequent research avenues have been explored – which you can investigate yourself through a literature search.
The main question Crawley and May (1987) asked was, “under what conditions do the annuals and perennials co-exist at equilibrium?”.
Note the proviso “at equilibrium”, this is to avoid considering the cases in which either of the species may hang around for a period of time but will eventually go extinct.
The perennials have two key parameters
1) The probability of invading a neighbouring empty cell, b (probability per plant per year)
2) The probability of dying in the course of a year, d (probability per plant per year)
The annuals have only one parameter
1) The number of seeds produced per year, c (seeds per plant per year)
Given these conditions we can then construct the model. We first of all need to decide on the ordering of events, Crawley and May (1987) chose
1) Perennials die with given probability
2) New perennial ramets are born into empty cells with a given probability (this includes cells occupied by annuals and cells where a perennial has just died).
3) New annuals germinate from seed
Note that there are other orderings that could be chosen and typically the decision is made to best represent the biology of the system. For example the death of perennials could still leave a site uninhabitable for a year or more.
Using this model Crawley and May found that co-existence was possible under certain circumstances, specifically as long as the seed production of the annual was sufficient to maintain the population in the few sites left uncolonised by perennials each year. Their analytical investigation, summarised below, sheds more light on why this is the case.
Insights from analytical predictions
Given the above rules it is possible to construct an analytical approximation for the population dynamics in the model. Conveniently the annuals have no effect on the dynamics f the perennials, making writing the equations easier.
1) At the start of the year, a proportion of the sites are occupied by perennials, P. For the remainder we will initially ignore the annuals (as they have no effect on the perennials) and say that the proportion of the sites that are empty, E is simply equal to 1-P . Note also that before colonisation, a proportion of the perennial sites will also become empty through perennial death – approximately d*P. This leaves on average the proportion of sites available for colonisation equal to E+d(1-E)=[E(1-d)+d]
2) The probability that an empty site will become colonised by a single neighbouring ramet is b. However, an empty site may become colonised by any of its 8 neighbouring cells in our model. Therefore the probability that a cell may become occupied by any of the cells is one minus the probability that it remains empty. The probability that it remains empty is (1-b)k where k are the number of the neighbouring cells that are occupied, so the probability of being colonised is 1-(1-b)k.
3) Using similar logic we can work out the probability that an empty cell will remain uncolonised by a new perennial ramet, which is [1-b*(1-d)*(1-E)]8. This is one minus the probability that a neighbouring cell colonises an empty cell to the power 8 (8 chances). The probability that a neighbouring cell colonises an empty cell iis the probability that a neighbour is occupied by a perennial (1-E) times the probability that the perennial neighbour doesn’t die in that generation (1-d) times the birth probability.
4) Given these two equations we can calculate a discrete time equation for the proportion of empty sites: Et+1=[Et(1-d)+d]*[1-b*(1-d)*(1- Et)]8, and Pt+1=1-Et+1.
5) The standard next step is to work out what the average proportion of sites occupied by perennials or empty cells is at equilibrium. This is found by setting Et+1=Et=E* and solving the equation above. The trouble is that the equation has expressions of E* right up to the power 8, which means that there are 8 possible values for E*. Only two of these are typically realistic E*=1, all sites empty at equilibrium, and 0<E*<1 meaning that there is a nonzero equilibrium proportion of sites occupied by perennials. For given values of b, d and c it is possible to numerically solve this equation and obtain the equilibrium abundances.
6) Another trick is to set Et+1=Et=E*=1 and perform a stability analysis of the equation above. This reveals that the perennial can persist (E*=1 is unstable) when b>d/(8*(1-d))
7) The authors then went on to calculate a similar discrete time equation for the annuals (which only depends on c and the proportion of unoccupied cells left after the perennials have colonised), and then used that to work out a critical birth rate needed to maintain the annual population (ignoring demographic stochasticity). This was c>1/ E*. So provided the seed production per annual is greater than the inverse of the average proportion of gaps then the annuals should persist.
This whole analysis assumes that the population is well mixed and that demographic stochasticity is zero. You will see from playing with the simulations that the analytic approximation doesn’t do very well in certain circumstances, specifically when the average occupancy of the sites by perennials is low. Can you figure out why the analytic approximation doesn’t do well in certain circumstances?
Crawley, M.J. & May, R.M. (1987). Population dynamics and plant community structure: competition between annuals and perennials. J. Theor. Biol., 125, 475-489.