Crocodile populations are size-structured, and for populations that are subject to harvesting, removal is typically size selective. For this reason, size-structured matrix models are typically used to analyse the dynamics of crocodile populations. The boundaries between the size classes used to classify individuals in these models are typically chosen arbitrarily. This is problematic because results can depend upon the number and width of size classes.
The recent development of continuous character population models termed integral projection models (IPM) has removed the need to arbitrarily classify individuals. These models are yet to be applied to harvested animal populations.
Using information obtained from the literature, we develop an IPM for crocodiles. We use perturbation analyses to investigate how altering size-specific demographic rates influences the population growth rate and the strength of selection on snout to vent length.
We find that perturbations can lead to complex responses. Sensitivity analysis to population growth and fertility selection reveals that the smallest animals and the sizes of early breeding individuals and their eggs may have more influence on these population biology parameters than previously thought.
Although our model is relatively simple, our results show that IPM can be used to gain theoretical insight into the possible consequences of altering size-specific demographic rates on the population and evolutionary ecology of harvested populations.
Integral projection models (IPMs) provide a powerful framework to analyse the dynamics of populations structured by continuous characters like body size (Easterling, Ellner & Dixon 2000; Ellner & Rees 2006; Rees & Ellner 2009). They are remarkably easy to construct from associations between a continuous character and (i) survival, (ii) development, (iii) inheritance and (iv) fertility. Once they have been constructed, IPMs, or high dimensional matrices that accurately approximate them (Easterling, Ellner & Dixon 2000), can be used to calculate numerous quantities of interest to population biologists. These quantities include ecological parameters like the population growth rate, evolutionary parameters including character heritability and life-history parameters like the mean and variance in lifetime reproduction. Numerical and analytical sensitivity analysis is a tool used to analyse IPMs and matrices, and sensitivities can be used to explore how altering a model parameter might influence ecological and evolutionary parameters (Coulson, Tuljapurkar & Childs 2010). A range of sensitivities of some of these quantities to model parameters have been developed for IPMs (Ellner & Rees 2006; Rees & Ellner 2009) and matrices (Caswell 2001, 2010) but little is known about the sensitivities of other quantities to model parameters. In this article, we develop an IPM for crocodiles, calculate multiple population biology parameters from it, and through the use of graphical analysis, examine how perturbing model parameters influence these quantities.
Prior to the development of IPMs, management strategies for populations structured by continuous characters were identified through the analysis of stage-structured matrix models assuming a few arbitrary body size classes. IPMs abrogate the need to arbitrarily classify individuals into wide body size classes – instead they treat the character as continuous. IPMs and classical low-dimensional stage-structured models can provide contrasting insights into the dynamics of populations (Easterling, Ellner & Dixon 2000). Consequently, it is potentially informative to construct IPMs for managed populations, especially if individuals within the population are selectively targeted for removal as a function of a continuous character. Nile crocodiles (Crocodylus niloticus) provide an example of such a species.
Unregulated commercial and recreational hunting of crocodilians prior to the 1970s depleted many populations, bringing some close to the brink of extinction (Ross 1998). Declines were primarily driven by the leather industry and trophy hunting. Since then, conservation concerns have raised environmental awareness and crocodile exploitation has changed to more sustainable methods (Webb 2002; MacGregor 2006; Abensperg-Traun 2009). Ironically this turnaround was caused by the same commercial incentives that provoked the initial overexploitation. The crocodilian leather industry supports a valuable global trade for the producer countries in excess of USD 50 million per year (Ross 1998; MacGregor 2002), and the rise in popularity of eco-tourism contributes significantly to the income of many countries (Telleria et al. 2008). Although a few populations remain endangered, many have successfully recovered (Ross 1998). The main current threat to crocodilians is loss of habitat although overexploitation remains a possibility in some areas. Crocodilians are a top predator in the wetland environment, and mismanagement could affect changes throughout the ecosystem (Mazzotti et al. 2009).
Body size is an important factor in crocodilian management for a number of reasons. For example, large individuals are more desirable for tourism (Telleria et al. 2008), trophy hunting (Lindsey, Roulet & Romanach 2007) and as breeding stock for crocodile farms, yet large adults cause the majority of human and livestock injuries and deaths (Aust et al. 2009; Dunham et al. 2010). In addition, small individuals and eggs are targeted by ranchers, and although the repatriation percentage may exceed the natural survival rate, it is not always implemented or enforced. Body size may be heritable in crocodiles, (Hanken & Wake 1993; Reed, Schindler & Waples 2011) with the body size of offspring partially resembling that of their parents. In such cases, selective phenotypic harvesting of animals can generate evolutionary and demographic changes within populations (Ratner & Lande 2001; Coltman et al. 2003; Bishop et al. 2009; Mysterud 2011). Altering the strength of selection on body size via size-selecting harvesting could already be affecting crocodilian populations (MacGregor 2002). Understanding the dynamics of body size in crocodilian populations consequently has a range of potential benefits, and a crocodile IPM consequently has potential to provide further insight into the dynamics of both harvested and unharvested crocodile populations.
We develop an IPM for Nile crocodiles to investigate the dynamics of the body size distribution. The model is a simplification of the real world, and factors such as environmental stochasticity or density dependence that would have an effect on a natural population of crocodiles are not included. This manuscript presents an initial step towards utilizing a novel modelling technique for crocodiles. Using numerical perturbation, we examine how altering parameters in the functions used to construct the model influence the structure of the model, and the following quantities that describe aspects of the dynamics of the body size distribution: population growth rate, mean lifetime reproduction, mean body size, mean annual recruitment and mean annual survival rates. Results are discussed with regard to current and future conservation management.
Materials and methods
Each individual has a variety of characteristics that can be measured, from genotypes to phenotypic traits. At a given point in time, a population level distribution of individual characters can be constructed. The character distribution may change if the environment changes, and, in harvested populations, if hunting pressure alters (Mysterud 2011; Reed, Schindler & Waples 2011). Perturbation-based approaches can be used to examine the consequences of such change on model predictions.
We construct an IPM (of the female component of the population) of snout to vent length (SVL), measured ventrally from the tip of the snout to the first scale row posterior to the cloaca. SVL is a standard measure of body size for reptiles. To parameterize an IPM, it is necessary to identify four demographic functions:
(1) SVL – survival function S(z, t)
(2) SVL – development function G(z′|z, t)
(3) SVL – inheritance function D(z′|z, t)
(4) SVL – fertility function R(z, t)
We parameterize a non-age-structured, deterministic IPM of SVL, the continuous character that we refer to as z. The model predicts the distribution of SVL at time t + 1, n(z′,t + 1), as a function of SVL at time t, n(z,t).
G(z′|z,t) describes the probability that an individual of size z at time t grows to size z′ at time t + 1. S(z,t) describes the probability of survival from t to time t + 1 as a function of SVL. D(z′|z,t) describes the probability that a reproducing female of size z at time t produces a recruit of size z′ at time t + 1 R(z,t) describes the number of hatchlings produced by a female that enter the population at time t + 1 (we assume a post-breeding census, where hatchlings emerge from eggs just prior to the population census and the fertility functions incorporates adult survival).
Data and functions were obtained by searching the ISI Web of Science for relevant literature concerning SVL, survival, offspring SVL and fertility of C. niloticus. We used linear and linearized models (for survival, inheritance and fertility) to identify parameter values (Fig. 1) incorporated into the IPM:
SVL – survival function S(z,t): we use the linearized function (SVL′t + 1 = −3·0 + 0·07 SVLt) parameterized by Bourquin (2007) that utilized 6 years of survival and recapture rates (Fig. 1a), the function is logistic in form:
SVL – development function G(z′|z,t): this describes ontogenetic development, how SVL at time t + 1 is related to SVL at time t. To estimate the development function, two bits of information are required: mean SVL at t + 1 given mean SVL z at time t and the variance around this association. To derive the function for mean crocodile development, we transformed age-SVL functions (Fig. 1b), to generate a time series of SVL at age a against SVL at age a + 1 (Fig. 1c) using data from a field (Games 1990; Maciejewski 2006; Bourquin 2007) and captive study (Hutton 1984). To complete the development kernel, it was necessary to estimate the variance in growth rates. We did not have information on repeated measures from individuals at different ages, which is really required to accurately estimate this variance. We gained an estimate using the residuals around the mean development function. Although this estimate is only approximate, sensitivity analysis revealed that our results were insensitive to this parameter. Our conclusions consequently hold despite the approximate nature of this parameter.
The development kernel was then constructed. If we define μ as the function describing the average growth (SVL′t + 1 = 17·8 + 0·89SVLt), and σ as the function describing the variance (SVL′t + 1 = 4·95 + 0·11 SVLt), we can write the equation below.
SVL – fertility function R(z,t): clutches of eggs were laid when the individual attained the mean minimum breeding size of SVL 120 cm (Appendix 1). The number of offspring produced was set as a constant 7·5 offspring per breeding female (Fig. 1d). This was derived from a mean of 45·5 eggs per clutch (per female) multiplied by the proportion of females in the population (0·54), the females that lay a clutch of eggs (0·66), the clutches that survive depredation (0·69) and proportion viable eggs (0·70) (Appendix 1 details the data and sources).
SVL – inheritance function D(z′|z,t): the relationship between parent and offspring SVL was estimated from a study by Maciejewski (2006) that described the SVL of recruits as 13·1 cm (n = 149) and the variance as 0·3 (Fig. 1e). Female size was not ascertained, and although literature does indicate a positive correlation between female size and egg mass (Thorbjarnarson 1996; Swanepoel, Ferguson & Perrin 2000), there are no data available to directly relate C. niloticus female size to hatchling size. We assumed a function SVL′t + 1 = 13·1 + 0·01 SVLt. The SVL – inheritance kernel (Easterling, Ellner & Dixon 2000) is described using similar logic as the development function with μ as the function describing the average development and σ as the function describing the variance:
The IPM is a simplification of the real world. We do not incorporate density dependence, stochasticity, spatial processes or age structure. Despite this, deterministic models such as ours can provide useful management advice (Caswell 2001) and have previously been used to guide crocodile management strategies (Tucker 1995). Insights from simple models often qualitatively agree with those achieved through the use of more complex models (Easterling, Ellner & Dixon 2000).
We approximated our IPM as a high dimensional matrix with 100 size classes. We examined how results changed as the matrix size was altered. Model predictions were broadly consistent across matrices of dimension 100, 200 and 300. The smallest size class contained individuals between an SVL of 13·1 and 15·5 cm, the size classes advanced with increments of 2·4 cm, the largest size class included individuals of SVL 247·6–250·0 cm. The maximum size of individuals was assumed as SVL 250 cm (c. 500 cm total length) – none of the population ever grew to this size in the model. From this model, we calculated the following demographic parameters using methods described in Coulson, Tuljapurkar & Childs (2010), Easterling, Ellner & Dixon (2000), Rees & Ellner (2009):
Lambda (λ) – the population growth rate at equilibrium.
Fertility selection differential (FS) – describes the difference in mean SVL between breeders and the rest of the population.
Viability selection differential (VS) – describes the difference in mean SVL across the entire population and those that go on to survive.
Mean annual survival across all individuals within the population.
Mean annual recruitment – the average number of recruits produced across all individuals within the population.
Mean SVL across all individuals within the population.
Mean lifetime reproductive success (R0) – the mean number of hatchlings produced by a female during their life span.
The statistical functions S(z, t), G(z′|z, t), D(z′|z, t), R(z, t) are functions consisting of an intercept, and a slope, which can be altered. To analyse our IPM, we independently perturb each parameter in each function by 10% and examine how each perturbation influences the population biology parameters we calculate. All parameters were perturbed upwards, so negative values were made less negative by multiplying by 0·90. To understand the consequences of altering a model parameter, it helps to consider the structure of the IPM further.
The matrix approximation of the IPM consists of a 100 by 100 block of numbers describing transition rates from each possible SVL class at time t to each possible class at t + 1. The transition rates are determined by the survival, development, inheritance and reproduction functions. We want to know how altering each function parameter changes each of the population biology parameters we calculate. To gain insight into this, we want to know how altering a model function parameter alters each transition rate within the matrix, as well as how altering each transition rate influences the population biology parameter we are interested in. We concentrate on l and FS to demonstrate how an ecological (λ) and evolutionary parameter (FS) change when function parameters are altered. Let us consider the population growth rate at equilibrium, λ and an arbitrary function parameter – called β. The change in the population growth rate that results from changing the parameter (whilst holding all other model parameters at their original values) is a partial derivative, which is written as .
When we change a parameter in a function, we change many transition rates aij within the matrix. For example, if we increase the survival function intercept, we increase survival rates of individuals across all SVL values. This means that multiple matrix elements change. To calculate , we consider two questions. First, how does altering β change each matrix element, and second, how does changing each matrix element influence λ?
The effect of altering a single matrix element on λ is a partial derivative and the effect of altering β on a single matrix element can be written If we multiply these two effects together, and take the sum over all matrix elements, we can then write (Caswell 2001):
To calculate these quantities we did three things:
First, we increase each aij by a small amount (0·1%) and examined how this influenced each of our population biology parameters, λ and FS. Second, we altered each model function parameter (by 10%) and examined how each aij was altered. Finally, we examined the sum – the overall effect on each population biology parameter of increasing each model function parameter. We will use these insights to comment on how different management strategies, that alter survival, development and fertility, are likely to impact the population biology quantities we calculate.
The original (unperturbed) transition matrix
We represent the matrix approximation of the IPM as a 3D histogram (Fig. 2a), where the x axis represents SVL at time t, the y axis SVL at time t + 1 and the z axis the transition rates between sizes. The diagonal ridge indicates the survival and growth of individuals from hatchlings (on the left) to mature individuals (on the right). The ridge along the top of the matrix indicates the number of recruits produced by females of increasing size. Hatchlings progress along the diagonal from left to right across the surface as they survive and grow to adulthood. The fertility ridge is much higher than the survival–growth diagonal as this is the number of recruits produced per breeding female distributed across a few hatchling size classes, whilst the diagonal represents survival and transition rates (between 0 and 1·0).
Comparison of the predicted Nile crocodile population with empirical evidence
The first step was to establish whether the model predicted population biology parameters that are in agreement with those obtained from field studies on C. niloticus. The stable age distribution was characterized by a high proportion of small crocodiles and few large animals (Fig. 2b), which is typical of crocodilian populations (Bourquin & Leslie 2011). The IPM predicted λ of 1·02, indicates a population increasing by 2% per annum – well below the maximum rate a crocodile population can increase, and in line with growth rates observed in the wild (Smith & Webb 1985; Craig, Gibson & Hutton 1992). This corresponded to a mean lifetime reproductive success of 2·32 offspring per female over her lifetime. Mean annual survival across all individuals was 0·25, and the mean recruitment rate was 0·77. We are unaware of estimates of R0, mean survival and recruitment rates from free-living populations. However, our results are consistent with results obtained from traditional stage-structured crocodilian models (Smith & Webb 1985; Craig, Gibson & Hutton 1992). The mean SVL of all crocodiles in the population was 33·5 cm, reflecting the large proportion of the population consisting of small individuals.
Sensitivity surface to λ and fertility selection
Sensitivity of λ and FS to matrix elements of the model showed contrasting patterns. λ was most sensitive to matrix elements that described growth and survival (Fig. 3a). Sensitivity indicated four small peaks for survival of the smallest crocodiles (hatchlings thorough yearlings). Subsequent survival and growth resulted in increasing sensitivity culminating in the largest peak for those of mid-size (SVL c. 120 cm, the size of early breeding animals) and a small peak in sensitivity for eggs. The strength of fertility selection was reduced for survival and growth rates amongst pre-reproductive adults, especially for hatchlings (SVL c. 25 cm). An increase in the survival and growth rates of early breeding size animals (and their eggs) increased the fertility selection differential (Fig. 3b). These results occur as perturbing different matrix elements alter the proportion of the population in different reproductive and non-reproductive size classes.
Effects of altering model functions on matrix approximation
The effect of a function parameter perturbation had varying affects across the transition rate matrix, combining both positive and negative value changes (Fig. 4). Increasing the intercept or slope of the survival function had a disproportionate increase in survival and growth rates of smaller individuals than to those that were larger. Increasing the development function (intercept and slope) caused increased and decreased transition rates across all size classes. Increasing the development intercept influenced growth rates to an equal extent across all sizes, whilst increasing the growth slope influenced larger individuals to a greater extent than those that were smaller. Increasing the number of offspring produced an expected increase in the top rows of the matrix, whilst increasing the intercept and slope of the inheritance function meant that larger hatchlings were produced. As with growth, some inheritance values combined positive and negative responses.
Product of the two partial derivative matrices
When the and surfaces were multiplied by the various surfaces, a new surface was generated. In Figs 5 and 6, we provide 3D histograms detailing the product of the two partial derivative matrices. Positive and negative effects occurred to λ and FS because of the perturbation of either the slope or intercept to a particular function. Increasing the survivorship intercept or slope both caused increases in sensitivity to λ for the smallest individuals with a concurrent decrease in sensitivity to FS for the same size crocodiles. Increasing the development intercept resulted in a rise in values that culminated in positive and negative influences to λ around the young breeding individuals. Fertility selection was strengthened for pre-breeding size animals with a reduction in values for young breeding individuals. The development slope, however, caused a reduction in values to l that became stronger as animals grew and survived to breeding ages. Inheritance perturbations caused fluctuating positive and negative responses amongst the eggs laid to young breeding females. Increasing the number of offspring caused a strong positive reaction amongst early breeding females to both λ and FS.
Sum of the product surfaces
Finally, we examined how the different population biology parameters were influenced by perturbations to each model parameter. We only report results where population biology parameters were substantially influenced (Fig. 7) – only survival and development intercepts and slopes influenced most of the population parameters substantially – and we only report perturbations to these functions. The direction of change in each population biology parameter was identical if intercepts (or slopes) of survival and development functions were perturbed. For example, increasing the survivorship intercept and slope functions both increased R0 although at different magnitudes. However, the effect of perturbing each function, or whether the intercept or slope was perturbed, differed across population biology parameters. An increase in the survivorship functions (intercept or slope) caused a decrease to FS, whilst a similar increase to the development function caused a considerable increase to FS. Altering the slope of the development function caused the most considerable overall effects (with the notable exception of mean survival that was most influenced by the survivorship intercept).
All populations consist of individuals, which can be measured on a number of continuous and discrete characters. At any point in time, a population can be characterized by a distribution of character values. This distribution can change with time as the environment fluctuates. All population biology parameters describe an aspect of a character distribution or its dynamics: some, like λ and the strength of selection, describe change in the character distribution over a time step; others, like R0, describe aspects of the dynamics of cohorts; whilst yet others, like the heritability, describe how character distributions correlate (or not) across distributions. It is not surprising that when models are constructed to include character distributions, it is possible to calculate numerous population biology parameters. IPMs describe the dynamics of character distributions, and recent work has begun to investigate how population biology parameters of interest to ecologists, life-history theorists and quantitative geneticists are related (Coulson, Tuljapurkar & Childs 2010). This article builds on this work by focus on a harvested species of crocodiles.
Current literature based on stage/age-based models intimate that crocodilian sensitivity to l is highest for individuals that have attained breeding size or have survived to the largest sizes and is lower for the egg and hatchling stages (Craig, Gibson & Hutton 1992; Tucker 2001). The original (unperturbed) model we present here partially supports this traditional view derived from broad stage/age-based model sensitivity analysis. However, the IPM reveals that the population growth rate is also sensitive to survival and growth rates of small individuals (SVL c. 25–50 cm) and young breeders (SVL c. 120–150 cm). This suggests that previous conclusions about the limited population growth consequences of selective removal of young crocodiles from the wild may have been overstated. One reason we observe the peaks in the sensitivity surface for λ is the multi-modal distribution of the stable size distribution (Fig. 2a and b). When individuals are grouped into broad categories, we group sizes that are common with those that are rare, averaging the sensitivity across a range of sizes. Our analyses reveal that such averaging can lead to conclusions that are inconsistent with those found from continuous-size models. Our results suggest caution in removing large numbers of smaller crocodiles from populations, as their selective targeting could depress population growth to a greater extent than previous assumed (Craig, Gibson & Hutton 1992; Tucker 2001).
Ours is the first analysis to examine how the strength of natural selection varies as transition rates are modified. To understand our results, it is helpful to consider in depth what a selection differential is. The fertility selection differential (FS) on SVL describes the difference in mean SVL across parents and mean SVL across the whole population. It is calculated as:
where z is the range of SVL values and definitions for other terms can be found in methods. When the development, survival and inheritance function are perturbed, size-specific fertility rates remain unchanged but the stable size distribution, n(z,t), is altered. When the recruitment function is perturbed, both R(z,t) and n(z,t) are altered. Such perturbations lead to a change in the density of individuals in each size class, including the proportion of the population in the breeding size classes. These changes translate into difference in mean SVL between breeders and the rest of the population.
Comparison of the sensitivity surfaces of λ and FS to matrix transition rates helps explain why different perturbations can generate the contrasting results we see in Fig. 6. If altering a function parameter increases growth and survival rates of small individuals, this negatively impacts the strength of fertility selection, but increases λ. In contrast, if a perturbation increases survival and transition rates of larger individuals, λ and the strength of fertility selection both increase.
There has been a tendency in conservation biology to use the partial derivative of a population biology parameter to a matrix element to inform management recommendations. Although such an interpretation should be treated with caution (Benton, Plaistow & Coulson 2006), the analysis of models to inform population management is a useful tool. In reality, though, it would rarely be possible to perturb a single matrix element. Environmental change and alterations in harvesting strategies can impact multiple matrix elements simultaneously. The chain rule (Caswell 2001) allows the consequences of altering multiple matrix elements at once on population biology parameters to be explored. This allows us to ask how our population parameters will change if the association between SVL and survival, fertility, development and inheritance changes.
The transition rate matrix responded to perturbations of survivorship and development functions along the diagonal of the matrix and the inheritance and offspring functions along the top edge. Perturbing the survivorship function parameters would intuitively raise the survivorship of individuals across the whole population. As the population structure is biased towards smaller individuals, the increased survivorship is most notable within the transition rates for the smaller individuals. Altering the development functions would shift the original function to allow all individuals to grow to larger sizes than they did before the matrix was perturbed. A consequence of the perturbations caused some of the transition rates displayed in the 3D histograms to show apparent simultaneous increases and decreases. In the matrix at time t (we shall call this point A), the perturbation has caused a positive effect which then moves on to cause a positive effect at time t + 1 (point B). As the perturbation passes (from point A to point B), the point A surface has moved from high to low, the decrease causing a negative value. The converse has happened at point B where the low to high effect has created an increase in value. This ‘wave effect’ continues across the entire matrix surface because of the nature of the perturbation affecting the whole population.
The overall effects to the sum of the product surfaces for λ because of perturbation of the survivorship function parameters both had positive influences on the smallest individuals that constituted the largest part of the population. An opposite reaction occurs for similar perturbations to FS, resulting in a negative influence for the same range of individuals. The increased survivorship would allow the younger individuals to play a more significant role in population growth yet have less of an influence on the future breeding animals. Conversely, increasing the development functions resulted in positive and negative influences to λ but increased the values of the early breeding females to FS that became important for maintaining a population containing large, healthy breeders. When the number of offspring is increased the eggs of the young breeding animals becomes more important to λ and FS.
Our results are consistent with previous findings (Coulson, Tuljapurkar & Childs 2010) that perturbing different parameters can have contrasting effects on different pairs of population biology parameters. For example, if the development slope is perturbed, population growth increases along with the strength of FS. In contrast, if the survivorship slope is perturbed, population growth increases whilst the strength of FS declines. Perturbing different parts of the demography can consequently have contrasting effects on population biology parameters like λ and evolutionary parameters used to estimate the strength of selection. Our results provide some insights into why.
The model we have constructed is, deliberately, a substantial simplification of the real world. We do not incorporate environmental stochasticity, density dependence or age structure, for example. We acknowledge that these are omissions of important factors that can have substantial influences on any population including crocodiles (Webb & Manolis 2001; Fukuda et al. 2011). This manuscript seeks to explore the use of IPMs within crocodilian management. Despite this, our results can be used to provide some insight into the consequences of harvesting crocodile populations. Harvesting will reduce survival rates, and depending whether it is selective or not may reduce survival rates in certain SVL classes rather than others. Our results show that reducing survival rates across all individuals would result in a decreased λ SVL and R0 but would increase mean recruitment, VS and FS. However, if survival rates were to target individuals of a specific size – perhaps small individuals for ranching or large individuals for trophy hunting – the effects on the population biology parameters we calculate would differ. For λ, the most severe repercussions would be seen because of perturbations of the hatchling and early breeding sizes, whilst FS would be most affected targeting the early breeding sizes. Both λ and FS could also be affected by perturbations to the very largest individuals.
By numerically perturbing our IPM, we use rather inelegant brute force to calculate sensitivities. Equivalent results could be obtained through analytical means using published (Caswell 2001) methods for λ. In time, it is likely that analytical expressions for the sensitivity of R0 and selection differentials will be derived, perhaps using the approach recently advocated by Caswell (2011). However, until then, we are limited by our mathematical abilities.
The IPM used in this article represents a general model of the Nile crocodile, parameterized from a range of published literature. Although we draw conclusions from the results, we acknowledge that this is a theoretical model and that further investigation into aspects such as inheritance and density dependence will provide a stronger foundation for management recommendations that would be implemented in the real world. We have followed the effect of perturbations to a range of demographic functions through the structure of an IPM and to the resultant predictions. Analysis of this IPM reveals the complex responses of a population to slight changes.
We would like to acknowledge Imperial College London and the National Environment Research Council (UK) for the Ph.D studentship grant. This article was only possible because of the data collection efforts of those cited to parameterize the model, many thanks. We thank the two anonymous reviewers for insightful comments that improved this article.
Fertility-related parameters and sources for Crocodylus niloticus used to derive the integral projection models body length – fertility function (z,t). Nest effort represents the proportion of breeding size females of the population that nest. Mean clutch size is the mean number of eggs laid in a single nest. Nest sites not depredated are the proportion of nests destroyed during the incubation period. Viable eggs represent the proportion of eggs that hatch a viable offspring. Min size breeding female is the minimum total length of a female recorded found to be capable of breeding.