Population characteristics are emergent properties of collections of individuals in ABMs as in the real world [24]. It follows that even if individuals are completely deterministic, it is not necessarily true that population characteristics are directly related to environmental characteristics such as weather year or habitat quality in simple one-to-one relationships. In particular, none of the population characteristics revealed in the present analyses were known in advance. Instead they emerged from characteristics of landscapes, weather and individual behaviour reported in field studies and programmed into the ABMs. Thus the revealed population characteristics require explanations. Ideally these will be in terms of the properties of individuals in simple mechanistic relationships, but there is no guarantee such relationships exist. Below, we suggest mechanisms in terms of known and programmed behavioural ecology, and we back up our suggestions where possible by simulation experiments [32] and further analysis of simulation data. However proof of the validity of our conjectures requires in some cases additional experimentation beyond the scope of the present paper.
There was considerable spatial variation in local carrying capacity (Figure. 1) as expected because habitat quality varied across landscapes, and variation between 500 × 500-m squares accounted for 23% – 30% of the variance in pgr in the general linear model (Table 1). It can be seen that carrying capacities broadly reflect the main features of the landscape shown in Figure. 1. Thus the primary habitats of the skylarks and spiders were the arable fields which occur in the triangular regions at lower right and top left, but there the voles and beetles were scarce. These distributions result from the known behavioural ecology of each species. Skylarks nest in arable fields because they prefer open and accessible low vegetation, and the modelled spiders are an opportunistic species specializing in disturbed habitats like arable fields. Voles prefer the permanent grass cover often found beside roadside verges and other linear features. The beetles prefer pasture and arable fields, but their numbers suffer in arable fields because of disturbance during farming operation [33].
Although carrying capacities varied, there were no differences between squares in the other key parameter, the return rate, except in the vole. This is shown by the parallelism of the pgr-density relationships, return rate being by definition the negative of the slope of this relationship. Invariance of return rates has also been reported in red kangaroos (Macropus rufus) in the pastoral zones in South Australia [18]. Invariance of return rates suggests, but does not prove, that the mechanisms of density dependence are the same in all squares.
Average return rates are shown in Table 2, and are less than one y-1 for all four species, indicating that a return to carrying capacity after disturbance takes more than a year. These estimates are likely overestimates because our method of estimation of return rates has an intrinsic bias towards one [34], so the true values may be lower than shown in Table 2. Return rates less than one indicate that populations are stable and show no tendency to oscillate about their equilibrium values [3, 35].
The reasons why average return rates are less than one differ between species. The simplest cases are the beetles and the spiders, where we conjecture that the local populations are not able to recover from disturbance within a single season because the season is short and the larvae suffer high mortality even at low density, due to their locally patchy distributions. Both beetles and spiders have high fecundity but spiders recover from low density faster and have higher return rates than beetles (Table 2) because their juvenile stages escape density-dependent mortality by aerial dispersal (ballooning) [11, 36]. By contrast beetle larvae do not disperse far [26] and so density remains patchy at small scales even in low-density years. The result is that beetle populations are slow to recover from low density and so have a lower return rate than spiders. Our interpretation here is supported by experimental manipulation of the dispersal characteristics of the beetle to resemble those of the spider, which resulted in the expected increase in return rate, to 0.85 +/- 0.01, y-1. The experiment only entailed increasing the maximum distance that beetle individuals could move in a day from 14 to 50 m, all other behavioural characteristics of the beetle were left unchanged. Return rates in voles are related to the size of within-square population fluctuations (Figure. 4). Return rates decline as the size of fluctuations increases (r3214 = -0.36, p < 0.001) from a value around one when fluctuations are small. The intermediary variable here is the size of the patch of habitat in which the voles live. In large patches adults compete for territories in contest competition and this results in a return rate of one, and little variation in numbers from year to year, because non-territorial animals remain within the patch moving in the interstices between territories. In small patches by contrast non-territorial animals are not able to hide and due to the large edge to area ratio must disperse outside the patch, where they usually die. This renders sub-populations in small patches vulnerable to extinction if the residents also die, after which population recovery is slow. Thus population fluctuations are higher and return rates are on average lower in smaller patches. Variation in return rates occurs in voles (Microtus arvalis) in Fennoscandia and eastern Europe [5, 15], and it would be interesting to know if this is associated as here with the size of local population fluctuations.
The importance of our results lies in our quantification of the effects of spatial and temporal heterogeneity on the population dynamics of the four study species. The magnitude of these effects has implications for how we understand and predict population dynamics in reality. The effects of spatial and temporal heterogeneity must be accounted for if we are to have accurate predictive models for use in management and conservation. The size of the temporal effects shown here also has implications for how we evaluate long term temporal change (e.g. through climate change), since where there are large temporal variations, longterm changes will not be discernible quickly.
The effects of spatial heterogeneity were here shown by a significant square-density interaction, seen here in the vole (Table 1). Such interaction terms should be included in autoregression analyses of spatially separated populations, where return rate is given by the first autoregressive coefficient [5, 6], but this has not been attempted to our knowledge except in [18]'s study of kangaroos. It is not possible to include all the interaction terms included here in analyses of real spatially separated populations, because there would not be enough degrees of freedom. In our simulations we overcame this by replicating the weather years. Nevertheless the interactions that were most important here could be included, these would be density*year and density*square. Together with the main effects of year and square this would allow analysis of whether both carrying capacities and return rates vary in space and time. This provides a method of answering the question as to whether there is spatial and temporal heterogeneity in the population's dynamics.
How general are our conclusions? There are several limitations to our study. The study species were selected because they represent different functional groups, and each has qualities that make them representative of other species, but more would certainly be better. Similarly other landscapes should be investigated. For example the landscapes experienced by northern Scandinavian voles are more homogeneous than those modelled here, so spatial heterogeneity should there affect population dynamics less. The effects of weather year are likely qualitatively robust, though quantitatively effects depend on the actual weather experienced. Lastly, we have not investigated the effects of population sizes in earlier years, i.e., Nt-1, Nt-2, but these would allow the identification of population cycles found in, e.g., some vole species [37, 38].
It may be questioned as to whether we are here studying reality, or just very complex models. It is important to stress that our ABMs are the best available representations of life in the study site, which is why we chose to work with them. So, the emergent population properties of the ABMs should provide the most accurate characterisation of the real populations that is currently possible. Accuracy is not however guaranteed and checking – and correction – will probably be needed over the foreseeable future. The detailed nature of ABM predictions allows many checks, and such testing is ongoing. Related to this, it may seem surprising that not all emergent population properties of the ABMs are fully understood. Obtaining these mechanistic explanations is part of our research programme, however this can be laborious in ABMs as in reality, and success is not certain, as explained at the start of this Discussion.