Density-dependent population dynamics and dispersal in heterogeneous metapopulations

Authors

  • Chloë M. J. Strevens,

    1. Mathematical Ecology Research Group, Department of Zoology, University of Oxford, South Parks Road, Oxford OX1 3PS, UK
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  • Michael B. Bonsall

    Corresponding author
    1. Mathematical Ecology Research Group, Department of Zoology, University of Oxford, South Parks Road, Oxford OX1 3PS, UK
    2. St Peter’s College, New Inn Hall Street, Oxford OX1 2DL, UK
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Correspondence author. E-mail: michael.bonsall@zoo.ox.ac.uk

Summary

1. Metapopulation microcosms were constructed to test the effect of four different types of habitat heterogeneity on the dynamics and dispersal in spatially extended systems; homogeneity, spatial heterogeneity, temporal heterogeneity and spatio-temporal heterogeneity. Resources were distributed across discrete habitat patches in bruchid beetle (Callosobruchus maculatus) metapopulations, and long-term time series were recorded.

2. Mathematical models were fitted to the long-term time series from the experimental systems using a maximum likelihood approach. Models were composed of separate birth, death, emigration and immigration terms all of which incorporated stochasticity drawn from different probability distributions. Models with density-dependent and density-independent birth, death and emigration terms were investigated and, in each case, the model that best described the empirical data was identified.

3. At the local scale, population sizes differed between patches depending on the type of heterogeneity. Larger populations were associated with higher resource availabilities. As a result of this, the variation between local population sizes was greatest when there was spatial heterogeneity in which mean resource abundance varied from patch to patch. Variation in population sizes within patches was largest when there was temporal heterogeneity.

4. Density-dependent processes leading to the regulation of local population dynamics in our experimental systems were strongest in homogeneity or temporal heterogeneity treatments. Associated with this, we found that these systems were best described using mathematical models with density dependence acting on mortality. In contrast, spatial and spatio-temporal time series were adequately described using density-independent population processes.

5. Experimental metapopulations showed varying degrees of density-dependent dispersal. Local net dispersal each week was primarily driven by the local population size and secondarily affected by neighbourhood population density. Mathematical population models illustrated the importance of explicit description of density-dependent dispersal in all systems except the homogeneous metapopulations.

Introduction

Landscape heterogeneity has a profound effect on metapopulation function by influencing both local population dynamics and the dispersal between patches. The metapopulation concept originally referred to homogeneous patchy landscapes in which dispersal was considered only in terms of the colonization of empty patches and was equally likely for all individuals in a population (Levins 1969). Since then metapopulation models have been greatly elaborated to include the wide range of heterogeneity found in natural landscapes (Fahrig 2007) and to incorporate different dispersal strategies (Hanski 2001; Clobert, Ims & Rousset 2004).

Landscape heterogeneity can occur through differences in the characteristics of the habitat both within and surrounding patches. Patches can vary in size (Hanski & Thomas 1994; Griffen & Drake 2008; Nielsen, Wakamiya & Nielsen 2008), quality (Anderson & Danielson 1997; Fahrig 2007; Griffen & Drake 2008; Jaquiery et al. 2008; Stasek, Bean & Crist 2008) and connectivity (Gulve 1994; Hanski 1994; Goodwin & Fahrig 2002; Benjamin, Cedric & Pablo 2008; Stasek, Bean & Crist 2008). All of these metrics affect both local and regional population dynamics. Temporal heterogeneity is an equally important aspect of landscape ecology as it describes the change in landscape structure over time because of stochastic and periodically variable environmental conditions. Both temporal and spatial heterogeneity affect the mechanisms of population regulation and hence the carrying capacities and population densities. The existence of a relationship between population density and population growth is well established (Nicholson 1933; Andrewartha & Birch 1954). Understanding the relative importance of density-dependent and density-independent processes (Turchin 1999; Brook & Bradshaw 2006), knowing how to describe the nature of these density-dependent processes and describing how and why they may differ (Murdoch 1994; Sibly et al. 2005; Bonenfant et al. 2009) are current research priorities in the field of population ecology.

Dispersal has a profound effect on the demographic and genetic structure of local populations within a metapopulation. As such, the importance of an explicit description of dispersal in metapopulation systems is widely recognized. The original metapopulation model (Levins 1969) considered only the colonization of empty patches and therefore ignored a great deal of the complexity of natural systems where all patches can lose and gain individuals through emigration and immigration respectively. Similarly, dispersal theory has developed from being a fixed trait where all individuals have an equal chance of dispersing to one where dispersal is conditional upon the population density in a patch, the age or sex of the individual, interspecific competition, matrix characteristics and patch geometry, and interpatch distances (Clobert, Ims & Rousset 2004). The evolution of dispersal in the presence of spatial and temporal heterogeneity has been the focus of a large body of work. In particular, it is suggested that temporal stochasticity encourages dispersal to a greater extent than spatial heterogeneity alone (Holt 1985; McPeek & Holt 1992). Many studies are based on the theory of ideal free distribution (Fretwell & Lucas 1970) where individuals should disperse to maximize their reproductive fitness (Holt 1985; McPeek & Holt 1992; Holt & Barfield 2001; Morris, Diffendorfer & Lundberg 2004). However, this theory assumes unhindered movement and habitat sampling by dispersing individuals which may make it inadequate to describe dispersal in many natural systems (Leturque & Rousset 2002). Rather, it is probable that dispersal is a function of local population dynamics, interspecific interactions and disturbance, patch size and isolation, and species-specific dispersal behaviour. In heterogeneous landscapes, the most obvious way in which local population dynamics influence immigration and emigration rates is through density-dependent dispersal. When an individual cannot sample and select habitats which maximizes its fitness, only emigration rates may be affected by density-dependent dispersal (Clobert et al. 2009). In this case, immigration is a function of the population densities in neighbouring patches within the network. In general, dispersal is positively related to density, with higher dispersal rates out of high-density patches where exploitative and interference competition is strong (Bowler & Benton 2005; Matthysen 2005). There is a reciprocal relationship between dispersal and population dynamics such that population dynamics affect dispersal which in turn influences population dynamics. Dispersal has been demonstrated to affect mean population densities, variation in densities and density dependence in local populations (Coffman, Nichols & Pollock 2001; Lecomte et al. 2004).

To test the hypothesis that the mechanisms regulating local population dynamics and dispersal differ depending on the nature of landscape heterogeneity, we use time series collected from metapopulations of the bruchid Callosobruchus maculatus. Metapopulation microcosms incorporating spatial, temporal and spatio-temporal heterogeneity, and homogeneity were constructed and replicated using a well-developed system (Bonsall, French & Hassell 2002; Bonsall & Hastings 2004; Bull et al. 2006; Hunt & Bonsall 2009). Time series of local population sizes and net dispersal were collected and analysed. These time series were then used to examine the relative importance of explicit descriptions of density-dependent and density-independent population dynamics, and dispersal using mathematical population models. Population models were composed of birth, death, emigration and immigration terms. The relevance of density dependence in all these terms, with the exception of immigration which was a constant based on neighbourhood densities, was examined by using a maximum likelihood approach to fit a range of density-dependent and density-independent models to experimental time series (Bonsall & Hastings 2004; Bonsall & Hassell 2005). Variation in birth, death, emigration and immigration terms comprising the population models were described using different probability distributions (Bonsall & Hastings 2004; Melbourne & Hastings 2008).

Materials and methods

This study combined an experimental and a theoretical approach to examine dispersal in heterogeneous patchy landscapes. Population dynamics were considered at both the local (patch) scale and the regional (metapopulation) scale.

Experimental system

Multi-generational metapopulations of C. maculatus were studied using experimental microcosms based on a system developed by Bonsall, French & Hassell (2002). Bruchid beetles (also known as Southern cowpea weevils), C. maculatus (Coleoptera: Chrysomeloidea: Bruchidae), are agricultural pests ranging throughout the tropical and subtropical world. Their simple, self-contained life history makes them ideal candidates for use in experimental model systems (Desharnais 2005). Each life cycle lasts c. 4 weeks though there can be substantial variation in adult longevity and larval development times. Adult bruchids live for 1–2 weeks during which time they neither eat nor drink but seek mates and oviposit on the surface of their bean resource. The adult phase is the only actively mobile life stage for this species. Larvae develop within the resource over the course of 3–4 weeks, nourished by the endosperm of the bean. When the metamorphosis from larva to adult is complete, the bruchid chews through the seed coat of the bean and becomes reproductively mature within 24–36 h.

Experimental landscapes consisted of nine transparent plastic boxes (73 × 73 × 30 mm) arranged in a 3 × 3 lattice (Fig. 1). Each of these boxes represented a patch within the metapopulation. Experiments were instigated by introducing two pairs of bruchids and their resources into each patch within the metapopulation each week for 3 weeks. By the fourth week, adults were emerging from the oldest beans and the populations became self-perpetuating. Thereafter, only beans were introduced each week. The resource used in this study was the black-eyed bean (Vigna unguiculata, Leguminosae). The number of beans introduced to each patch was dependent upon the type of heterogeneity in the landscape. Four heterogeneity treatments were implemented, each of which was replicated four times:

Figure 1.

 Schematic of the 3 × 3 metapopulation lattice used in this study. This illustrates the fraction of dispersers immigrating into a focal patch from neighbouring patches in the lattice. Here, a third (4/12) of all connections lead into the central patch (grey) from neighbouring patches. The same technique was used to calculate the fraction of dispersers entering corner (dotted) and edge (striped) patches.

  • 1 Homogeneity: two beans were introduced to all patches each week for 50 weeks. This meant that 18 beans (two beans × nine patches) were added to the metapopulation each week and 900 (18 × 50 weeks) beans were introduced regionally by the end of the experiment.
  • 2 Spatial heterogeneity: the number of beans added to a patch each week was drawn from a Poisson distribution with a mean of two. The total number of beans added to the system any week was constrained to equal 18. The same number of beans was introduced to patches every week.
  • 3 Temporal heterogeneity: a mean of two beans were added to each patch per week with variation across time drawn from a random number generator based on a Poisson distribution. The total number of beans introduced to the entire metapopulation over the course of the experiment was 900.
  • 4 Spatio-temporal heterogeneity: resources were distributed randomly across both space (in the same way as treatment 2) and time. Temporal heterogeneity was imposed in the same way as in treatment three except that the Poisson distribution used had a different mean depending on the patch and the mean number of resources it contained. The same total number of beans was introduced to the system as in the three other treatments.

Dispersal throughout the system was facilitated via 50-mm-long, 4·5-mm-wide tubes connecting neighbouring patch faces. To encourage asynchrony between local patch dynamics, dispersal was limited to 2 h per week. Prior to dispersal each week all living individuals were counted, dead individuals were removed and the oldest beans replaced with new ones. Old, removed beans were stored for an additional 4 weeks to collect any late emergents and these were reintroduced to their natal patches. A second population count was performed after dispersal and compared to the pre-dispersal count. Long-term time series of local population sizes and net dispersal were recorded.

In this single-species system, bruchids are both locally persistent and have high dispersal rates. As a result, this metapopulation may be more aptly described as a well-connected patchy population than as a classical metapopulation in which local extinction events are relatively frequent (Harrison & Taylor 1997).

Statistical analysis

Population and net dispersal time series were compared between treatments using linear mixed effects models which distinguish between random effects (replicate) and fixed effects (treatment) in the explanatory variables.

Density dependence is most commonly assessed by regressing a demographic rate against a measure of population density or size (Hassell 1975; Royama 1992; Bonenfant et al. 2009). Here, given the continuous population dynamics in the system, density dependence was estimated as the slope of the line relating the population growth rate (Rt = ln(Nt+1/Nt)) to log population size (ln(Nt)), where each time step corresponds to the weekly census. This provided a single measure of the way the population size changed from one time step to the next across the whole time series.

Both a linear model and a generalized additive model (GAM) were fitted to the data describing the relationship between net dispersal and log local population size and neighbourhood population size across all treatments. Here, neighbourhood population size was the sum of the local population sizes in all neighbouring patches. The best-fit model was selected based on the amount of variation in the data and the residuals it explained.

A measure of the strength of density-dependent dispersal was calculated from the slope of the line relating log local population size and net dispersal. The effect of resource abundance, heterogeneity treatment and patch connectivity on the strength of density-dependent dispersal was examined using a linear mixed effects model. Patches differed in their connectivity indices depending on the number of dispersal faces shared with neighbouring patches. Specifically, corner patches had two connections, the central patch in the lattice had four connections and all other patches had three connections (Fig. 1).

Population modelling

Population dynamics in the experimental systems were further examined using a set of candidate population models. This allowed us to estimate the importance of several explicit descriptions of density dependence in populations under the different heterogeneity treatments.

Local populations grow because of births and immigration and decline because of deaths and emigration. Density dependence was introduced to the birth (f(N)), death (g(N)) and emigration (h(N)) terms using two functions taken from Bellows (1981). The first function was a one parameter function from Skellam (1951) with density dependence acting on births represented by (1 + aN)−1 and density dependence on deaths and dispersal described by ln(1 + aN). The second function was a two parameter function introduced by Hassell (1975) where density dependence on births is described by (1 + aN)b and that on deaths and dispersal is described by b(ln(1 + aN). Logistic population growth models with density-(in)dependent dispersal were also considered. A full list of the population models studied is given in the Supporting Information (Table S1).

To initiate the model fitting, estimates for density-independent birth rates (r), death rates (d) and constants defining the strength of density dependence on births and deaths (abirth, bbirth, adeath, bdeath respectively) were made from the time series data for births and deaths. The density-independent emigration rate μ and the strength of density-dependent dispersal (adisp, bdisp) were approximated based on the relationship between population size and negative net dispersal (numbers leaving a patch each week). The number of immigrants (λ) was estimated from:

image(eqn 1)

where μ is the emigration rate and Nneighbourhood is the total number of bruchids in neighbouring patches. The number of dispersers out of neighbouring patches (μNneighbourhood) is multiplied by the average probability of dispersing into a focal patch (1/3). The probability of dispersing into a focal patch was calculated from the number of connections between neighbouring patches and a focal patch as a proportion of the total number of connections from neighbouring patches (Fig. 1). There was a marginal difference in the probability of dispersing into a focal patch depending on its position in the lattice. Patches in the corner and centre of the lattice had probability value of 1/3 whilst edge patches had a probability value of 3/8. Owing to the small difference between these values, we approximate the probability of dispersing into any patch within the lattice as 1/3.

Models were fitted to all replicates of all local population time series for each treatment using a maximum-likelihood-based approach where the likelihood (L) is a value of an unknown parameter set (P) from an empirical data set (D) generated by a particular model (M) and can be expressed as:

image(eqn 2)

Variation in births, deaths, emigration and immigration was described using different distributions (Bonsall & Hassell 2005; Melbourne & Hastings 2008). Parameter estimates for births and emigration were based on local population time series and as such were described by the same distribution. Separate distributions described the variation in deaths which were based on mortality time series and in immigration that was based on neighbourhood population time series. A negative binomial distribution was used to describe the demographic variability in births (and emigration) and deaths:

image(eqn 3)

where ni and Ni are observed and expected values for bruchid abundance at the ith time point respectively, N* is the mean abundance in the time series, k is a shape parameter and г is the gamma function ((x−1)!). The first term of the expression inline image refers to the first point in the time series n1 conditioned on the mean N*. The second term inline image relates to subsequent values in the time series conditioned on Nt−1. Although the Poisson distribution has been used in similar studies to describe demographic stochasticity (Bonsall & Hastings 2004), it was inadequate given the extent of over-dispersion in the data in this study.

Immigration was described using a Normal distribution. This allowed us to describe a mean level of dispersal into each patch (associated with immigration being based on neighbourhood patch dynamics) and is a straightforward way to describe the stochastic variation in the data (eqn 4):

image(eqn 4)

where σ2 is the variance, N* is the mean abundance and ni is the observed abundance at the ith time point. The negative log likelihood of the unknown parameters was minimized using an appropriate optimization regime (Nelder & Mead 1965). The best-fitting model was then selected based on Akaike Information Criteria (AIC) scores (Akaike 1974) and weights (Burnham & Anderson 2002).

Local population time series for each treatment were then simulated based upon their best-fit models. Both deterministic and stochastic simulations were run. For the latter, the likelihood of population increases because of births or immigration or decreases because of deaths or emigration at any point in time was a stochastic process (Renshaw 1993). In single patches, the population model was decomposed into four functions; E1 (birth), E2 (death), E3 (emigration) and E4 (immigration). The transition probabilities, T, for each of these events are described as:

image(eqn 5)

This allowed the probability of an event within a small interval of time (TTT) to be divided into four processes: birth (E1/T), death ((E+ E2)/T), emigration ((E+ E+ E3)/T) and immigration ((E+ E+ E+ E4)/T. To determine which of these events took place at any point in time, random numbers were generated from a uniform distribution using a method from Press et al. (1992). Depending on the size of the random number generated relative to the transition probabilities of E1–4, the population was either increased (birth and immigration) or decreased (death and emigration).

To interrogate the model fits one step ahead plots were used. These calculate the population size at time + 1 based on the deterministic model acting on the mean local population sizes at time t. In doing this, they can capture some of the variation in the local population sizes. One step ahead predictions were compared to the mean local time series from our experiments and the mean local time series from the stochastic population models.

Results

Experimental results

Regional and local time series for each of the four heterogeneity treatments are illustrated in Figs 2a–d and 3a–d, respectively. There was a significant difference between local population sizes depending on the type of heterogeneity in the landscape (F3,7613 =71·89, < 0·001). There was a larger difference between patch sizes in spatially heterogeneous systems. This is because of a strong positive relationship between resource abundance and population size, with resource-rich patches supporting larger populations (F10,7032 =690·95, < 0·001). However, there was no difference between regional metapopulation sizes between the different treatments (F3,841 =0·157, = 0·925). Variation within local populations also differed significantly between treatments (F3,137 =1437·02, < 0·001). In particular, there was more variation in population sizes as a result of temporal and spatio-temporal heterogeneity.

Figure 2.

 Regional population time series from each of the four replications of the experimental treatments; (a) homogeneity, (b) spatial heterogeneity, (c) temporal heterogeneity, (d) spatio-temporal heterogeneity.

Figure 3.

 Local population time series for one replicate of each experimental treatment; (a) homogeneity, (b) spatial heterogeneity, (c) temporal heterogeneity, (d) spatio-temporal heterogeneity.

Density-dependent population growth measured at the local scale was strongest in control and temporal treatments, coinciding with less unexplained variation by the linear models (Fig. 4). All treatments exhibited negative density dependence.

Figure 4.

 Density dependence in the populations was characterized using the relationship between the log(population size) and the log(growth rate) where growth rate is calculated from (ln(Nt+1/Nt)). The slope of the lines and the amount of variation explained by the linear models are noted on each panel. These show weaker negative density dependence and greater variation in systems with spatial and spatio-temporal heterogeneity.

The amount and direction of dispersal varied between patches within metapopulations depending on the type of treatment (F35,7596 =12·07, < 0·001). Variation was greatest in those systems with a spatial heterogeneity component (Fig. S1, Supporting Information). This was a result of the uneven distribution of resources. It is interesting to note that all patches, including those which had the smallest amount of resources available, showed both positive dispersal and negative dispersal at different points in time.

The relationship between net dispersal and local and neighbourhood population size was explored using a linear and a general additive model. Both models were able to explain the variation in the relationship between the log local population sizes and dispersal (linear model R= 0·51, GAM R= 0·66) but were poor at explaining the variation between neighbourhood population size and dispersal (linear model R= 0·02, GAM R= 0·03). When both log local and neighbourhood population sizes were included in the model, the generalized additive model was better able to describe the deviation in the data and as such it was used to examine the relative importance of these two variables (linear model R= 0·53, GAM R= 0·69). Using a generalized additive model, we found a significant interaction between net dispersal and local and neighbourhood population sizes (= 84·057, < 0·001). There was a positive interaction between local population size and dispersal, with larger amounts of dispersal out of patches with increasing population size (Fig. 5). When neighbourhood population size was larger, immigration into a focal patch was increased. Using the simple linear relationship between net dispersal and population size, we found that the strength of density-dependent dispersal was significantly affected by both heterogeneity treatment (F3,130 =3·20, < 0·05) and resource abundance (F5,130 =106·17, < 0·001) but found no significant effect of patch connectivity (F2,130 =1·56, < 0·21). Unsurprisingly, higher resource abundance resulted in stronger density-dependent dispersal. Whilst the mean and median values for density dependence was similar in all treatments, there was substantially more variation in density dependence in temporal and spatio-temporal systems.

Figure 5.

 Local populations displayed density-dependent dispersal. A generalized additive model was used to fit a curve to the relationship between the log(local population size) and local net dispersal. Negative net dispersal values indicate a net loss of individuals (emigration) and positive net dispersal values indicate net gain of individuals (immigration).

Model results

A full list of the population models studied, their AIC differences and weights for each treatment are listed in Table S1 in the Supporting Information. There was only support for one model to describe the population dynamics in each experimental treatment (Table 1). Homogeneous metapopulations were best described using a birth-death model with the single parameter density dependence function (Skellam 1951) acting on the mortality term and density-independent dispersal. Spatial and spatio-temporal systems shared a best-fit model in which population dynamics were density-independent and dispersal was density-dependent according to the two parameter density-dependent function (Hassell 1975). Finally, the best-fit model for temporal heterogeneity was one containing density-dependent mortality based on the two parameter function (Hassell 1975) and density-dependent dispersal characterized by the single parameter model (Skellam 1951).

Table 1.   Models with various combinations of density dependence acting on births, deaths and dispersal were fitted to the experimental time series (Table S1, Supporting Information). The model parameter estimates for the best-fitting models with approximate 95% confidence limits (Morgan 2000) are given
TreatmentBest-fit modelParameter estimates
  1. r is the density-independent growth rate, d is the density-independent mortality rate, μ is the density-independent emigration rate, λ is the number of immigrants each time step, v is the variance in immigration described using a normal likelihood distribution, kbirth and kdeath are the variation parameters for birth and death rates respectively using a negative binomial likelihood distribution, adeath, bdeath, adisp, bdisp describe the strength and nature of density dependence acting on deaths and dispersal rates respectively.

HomogeneityDensity-dependent mortality (Skellam 1951)
Density-independent dispersal
dN/dt = rN − Nln(1 + adeathN− μN + λ
= 0·14 ± 0·02, adeath = 0·01 ± 0·00, μ = 0·13 ± 0·01, λ = 2·11 ± 0·02, = 2·41 ± 0·00, kbirth 5·45 ± 0·02, kdeath 4·69 ± 0·01
Spatial heterogeneityDensity-independent population dynamics
Density-dependent dispersal (Hassell 1975)
dN/dt = rN − dN − bdispNln(1 + adispN) + λ
= 0·58 ± 0·01, = 0·61 ± 0·01, bdisp = 2·16 ± 0·01, adisp = 0·02 ± 0·00, λ = 4·93 ± 0·01, = 2·84 ± 0·00, μ = 0·63 ± 0·00, kbirth 2·20 ± 0·02, kdeath 1·08 ± 0·01
Temporal heterogeneityDensity-dependent mortality (Hassell 1975)
Density-dependent dispersal (Skellam 1951)
dN/dt = rN − bdeathNln(1 + adeathN)−Nln(1 + adispN+ λ
= 0·58 ± 0·01, bdeath 0·44 ± 0·01, adeath 0·09 ± 0·00, adisp 0·03 ± 0·01, λ 2·94 ± 0·00, = 2·35 ± 0·01, μ 0·18 ± 0·00, kbirth 1·89 ± 0·00, kdeath 1·80 ± 0·01
Spatio-temporal heterogeneityDensity-independent population dynamics
Density-dependent dispersal (Hassell 1975)
dN/dt = rN − dN − bdispNln(1 + adispN) + λ
= 0·62 ± 0·00, = 0·77 ± 0·00, bdisp 0·86 ± 0·02, adisp 0·17 ± 0·00, λ 5·04 ± 0·01, = 3·16 ± 0·00, μ 0·67 ± 0·00, kbirth 2·78 ± 0·01, kdeath 2·29 ± 0·01

The goodness-of-fit of these models was investigated using the ability of simulated time series to describe observed population dynamics. As the models were fitted to all local time series, the simulations were compared to the mean local dynamics for each treatment (Figs 6 and 7). This shows that the best-fit deterministic model provides a good description of the mean dynamics in the homogeneous treatment. However, the models tended to overestimate the mean population sizes in the spatially heterogeneous systems and underestimate those in the temporally heterogeneous systems. In contrast, stochastic models described the dynamics in the spatially heterogeneous systems well but tended to overestimate the dynamics in the homogeneous and temporally heterogeneous systems and underestimate the dynamics in the spatio-temporal treatment. These disparities are likely due to the fact that the simulations failed to capture the high degree of demographic variability acting on local populations. Furthermore, models were fitted to all local populations from each treatment at once and did not separate patches according to local resource abundance. This does not allow for the variability in population dynamics between patches where population sizes varied significantly.

Figure 6.

 Time series illustrating the simulated dynamics from the deterministic model (thick line) plotted on the mean local dynamics from the experimental microcosms with standard errors. This shows that the models (Table 1) adequately described the population dynamics in homogeneous systems but either over-estimated (spatial and spatio-temporal heterogeneity) or underestimated (temporal heterogeneity) the population sizes in heterogeneous systems.

Figure 7.

 Time series illustrating the simulated mean local dynamics from ten replicates of the stochastic model (thick line) plotted on the mean local dynamics from the experimental microcosms with standard errors. Model simulations were adequate to describe the mean population dynamics in spatially heterogeneous systems, but either over-estimated (homogeneity, temporal heterogeneity) or under-estimated (spatio-temporal heterogeneity) the mean population sizes in the other treatments.

One step ahead predictions were compared to the mean local time series from our experiments and the mean local stochastic population simulations (Fig. 8). One step ahead predictions show a consistent relationship with observed mean dynamics (Fig. 8a). The relationship is more diffuse with the simulated dynamics because of the inability of the models to describe demographic stochasticity acting on births, deaths and emigration (Fig. 8b). The variation explained by the one step ahead predictions was also greater for the observed time series than the estimated time series for the same reason (Fig. 8c,d). Graphs of the residuals plotted against time show no trend in the variability over time (Fig. 8e).

Figure 8.

 Graphs of the one step ahead predictions (osap) against (by row) (a) the experimental time series (obs) and (b) the deterministic model time series (est) for each treatment. The variation explained by the models is shown in the residual plots (osap resid) against the experimental (obs) and simulated (est) time series (c and d respectively). Finally, any temporal trends in the models are shown in the plots of the residuals against time (e).

Discussion

Here, we have examined the nature of regional and local population dynamics and local dispersal in experimental heterogeneous metapopulations. Four types of landscape structure were studied; homogeneous, spatial heterogeneous, temporal heterogeneous and spatio-temporal heterogeneous habitats. In all cases, heterogeneity was a product of resource distribution across the landscape. We compared the results with population models containing various types of density-dependent and density-independent birth, death and dispersal processes which were fitted to the data.

Spatial and temporal landscape heterogeneity affects regional and local metapopulation dynamics in different ways. In general, resource abundance is positively correlated with population size and growth rate and, as a result, patches in spatially heterogeneous systems tend to have different carrying capacities (Andrewartha & Birch 1954; Fahrig 2007). Our results agree with this, showing a significant difference in population sizes at the local scale depending on the different underlying heterogeneity treatment. In particular, there was a greater variation between local population sizes in metapopulations with spatial and spatio-temporal heterogeneity, with a strong positive relationship between resource abundance and local population size. The difference between patches was balanced out at the regional scale where we found no difference between heterogeneity treatments. Temporal heterogeneity had the obvious effect of increasing the variation in population sizes within patches. The type of heterogeneity also had a significant effect on the strength of density dependence acting in local populations. In all cases, density dependence was negative, but it was weaker in systems with spatial and spatio-temporal heterogeneity. In their study of ungulates in the Rocky Mountains, USA, Wang et al. (2006) showed that temporal heterogeneity strengthened density-dependent population regulation whilst spatial heterogeneity weakened it. Here, we have a similar result where weaker density dependence was exhibited in experimental landscapes where there was spatial and spatio-temporal heterogeneity. Using a numerical integration process to minimize log-likelihoods and find models which best described our time series data, we determined the importance of explicit descriptions of density dependence acting on local births and deaths. These showed that explicit descriptions of density dependence were only necessary in the case of homogeneity and temporal heterogeneity. Both spatial heterogeneous systems were best described by models with density-independent population dynamics. This trend may be because increased levels of dispersal arising from spatial heterogeneity which could reduce the competition for resources at the local scale in these systems.

Differences between local carrying capacities result in asymmetrical dispersal between patches when dispersal is density-dependent. This establishes the foundations for the field of source-sink metapopulation ecology whereby sources have larger populations resulting in higher emigration rates and the opposite is true of sinks (Pulliam 1988; Pulliam & Danielson 1991). In source-sink systems, the net flow of dispersal is from sources to sinks. In this study, we found variation in net dispersal in patches depending on the type of heterogeneity present in the systems with the greatest variation between patches occurring in systems with spatial heterogeneity. Analysis of the relationship between dispersal and density revealed positive density-dependent dispersal in which larger local populations resulted in larger net losses of individuals through dispersal. Positive density-dependent dispersal because of increased interference and exploitative competition in large populations has been recognized in many previous studies (e.g. Bowler & Benton 2005; Matthysen 2005). Mathematical modelling of the local population time series from our experimental metapopulations illustrated that explicit descriptions of density-dependent dispersal were necessary to describe population dynamics in all the heterogeneous landscapes studied. In contrast, homogeneous metapopulations were best described using a density-independent dispersal process. We propose that this occurs as dispersal between patches in homogeneous systems may be balanced out across the network.

The net flow of individuals between patches because of density-dependent dispersal can be considered in terms of an ideal free distribution (Fretwell & Lucas 1970; McPeek & Holt 1992; Holt & Barfield 2001). For this process to occur, individuals must be able to sample and choose the habitat which will maximize their reproductive fitness. The reciprocity of movement between patches is then directly related to the population growth rates in patches compared to one another at any point in time (Morris & Diffendorfer 2004; Morris, Diffendorfer & Lundberg 2004). This would be an interesting extension to the work outlined in this study. However, alternative approaches than those adopted here are necessary to examine this phenomenon as dispersal was only measured as a net loss or gain of individuals at each time step so that the natal patch of each disperser was unknown.

Whilst local population density was the primary driver of dispersal in this study, there was a secondary effect of neighbourhood population density. Here, we found that larger neighbourhood population sizes resulted in increasing immigration into focal patches. This illustrates the importance of neighbouring patch quality and population dynamics in defining the dynamics of a focal population. The effect of the source-sink character of neighbouring patches has been well studied. Nearby high-quality sources can bolster or rescue extinction-prone patches (Brown & Kodric-Brown 1977). Sources may also increase metapopulation persistence because larger populations tend to be more persistent and less prone to extinction because of demographic stochasticity (Lande 1993).

Mathematical models have been widely used to describe and predict metapopulation dynamics (Gilpin & Hanski 1991; Desharnais 2005). We have used models to examine the nature of density-dependent population dynamics and dispersal in the experimental time series. Our basic model was comprised of separate birth, death, emigration and immigration terms and was fitted to data using a maximum likelihood process. Stochasticity was included in each of the terms separately, based upon different probability distributions (Lande, Engen & Saether 2003; Bonsall & Hastings 2004; Melbourne & Hastings 2008). We found that the most appropriate distribution to describe demographic stochasticity was a negative binomial distribution as the Poisson distribution was unable to capture the over-dispersion, likely due to demographic variability, in the time series (Melbourne & Hastings 2008).

Using model simulations to study the ability of the model to describe observed population dynamics, we found that both stochastic and deterministic simulations failed to account fully for the variability in heterogeneous metapopulations. In contrast, the deterministic model provided a good description of population dynamics in the homogeneous metapopulations. The disparity between simulated and observed population dynamics may be because demographic stochasticity was not explicitly described in the model simulations. Demographic stochasticity was introduced during the model fitting phase using negative binomial likelihood distributions for births, deaths and emigration. However, in the model simulations, stochasticity was based on a Poisson process which assumes that there is no heterogeneity in birth or death rates. To incorporate the effects of demographic stochasticity more accurately, separate probability distributions should be applied during the stochastic simulation process as was done in the model fitting phase. Furthermore, models were fitted to all local population time series at once, rather than separating them out according to local resource abundance. As a result, all patches in the model simulations were described by the same model parameters. This would decrease the accuracy of the model for heterogeneous systems. Upon interrogating the fit of these models to the experimental time series using one step ahead predictions, we showed that the deterministic models can appropriately describe the observed population dynamics when the variation in the mean local population sizes is incorporated.

In conclusion, this study illustrated that different types of landscape heterogeneity have dissimilar effects on population dynamics and dispersal in metapopulations. A comprehensive knowledge of the regulatory processes and their mechanistic manifestations underlying metapopulation dynamics is essential to manage and conserve natural fragmented landscapes.

Acknowledgements

We are grateful to Nina Alphey and Brian Strevens for their advice and support and to two anonymous reviewers for their comments on this work. This study was supported by the Natural Environment Research Council (NER/S/A/2006/14181), a scholarship from Magdalen College Oxford (CMJS) and The Royal Society (MBB).

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