Life cycle trade-offs in matrix population models

Ecology, Dec, 1995 by Peter H. Van Tienderen

INTRODUCTION

The analysis of the dynamics of age- or size-structured populations by matrix projections is acquiring a prominent place in ecology (Van Groenendael et al. 1988, Caswell 1989). A projection matrix contains matrix elements that correspond to survival of individuals, transition to a next stage or age class, or fecundity. The age or stage distribution after one time interval, often 1 yr, is then calculated as the product of the projection matrix and the vector with the current numbers of individuals in the age or stage classes. Most analyses concern the identification of those life cycle components (summarized by the matrix elements) with a high potential impact on population growth (e.g., Crouse et al. 1987, Kalisz and McPeek 1992, Silvertown et al. 1993). This impact is measured as either the sensitivity or elasticity of the respective matrix elements (Caswell 1978, 1989). Sensitivities and elasticities are scaled differently. The sensitivity of a transition parameter can be used to evaluate the effect of a small absolute change in the parameter on the population growth rate. Matrix elements often have different units of measurement and therefore sensitivities cannot be used to compare the effects of different elements. For instance, it is not meaningful to compare the sensitivities for survival and seed production, because of their different dimensions. In contrast, elasticities are dimensionless: they give the proportional increase in population growth rate per proportional increase in a matrix element. This facilitates the comparison of the effects of different matrix elements. However, a high proportional change in a matrix element may correspond to a small change in absolute value, and vice versa.

Sensitivities and elasticities have in common that they evaluate the consequences of variation in a single matrix entry only. The effects of within-population variation and covariation in matrix elements have received relatively little attention. Yet exactly this aspect may be important biologically. Without variation, the potential effect of a matrix element on population growth is not clear. Also, the net effect of a focal projection matrix element on the population growth rate may be smaller than its sensitivity or elasticity, or even negative, due to negative correlations with other elements. For many biologically relevant questions it is important to take this covariation into account. Trade-offs among components of fitness, such as survival and fecundity, are central to life history theory (Stearns 1992). Trade-offs may arise due to pleiotropic effects of genes that affect life cycle components (Templeton 1980). When matrix entries comprise life cycle components that are involved in a trade-off relation, a change in one matrix element may be associated with a change in the opposite direction in another element. For instance, current reproduction may be negatively correlated with future survival. Then, the net consequences of a shift in reproduction cannot be inferred from single entry sensitivities or elasticities. Alternatively, matrix elements may be coupled to other elements merely by the fact that they represent mutually exclusive options; for instance, individual juveniles in a juvenile stage class may remain in that class, become reproductive adults, or die. This may lead to negative correlations among matrix entries, and the importance for population growth of a change in such transitions can only be assessed by taking these unavoidable associations into account. Finally, perturbations of the life cycle (for instance by protective measures of endangered species, or by environmental changes) may affect several matrix elements; in such cases it becomes important to integrate all effects on the population growth rate.

In order to evaluate the effects of correlations among matrix elements, a connection is needed between the properties of the projection matrix as measured by sensitivities and elasticities, and information on the (co)variation of the elements of the matrix. The main problem addressed is: how can variation in matrix elements and the covariation among them be accounted for in a matrix projection analysis? Here, the use of integrated sensitivities and elasticities is advocated; they measure the total effect of matrix elements on the population growth rate, separating direct effects from indirect effects through correlation with other entries. The underlying theory is rather dispersed (Charlesworth 1980, Templeton 1980, Caswell 1983, 1985, 1989, Lande 1982a, b), one reason why applications possibly are scarce. This paper aims to popularize viable approaches and extend their usage, rather than to derive new results. First, the general procedure is described that allows adding information on (co)variation among transition parameters to matrix projection models. Then, the interpretation of integrated measures depending on the source of the covariation (due to sampling errors, phenotypic, or genetic variation) is addressed, and some of the limitations of the analysis are discussed. Finally, two examples are worked out in more detail. It is shown that in the populations of Plantago lanceolata infested with the fungus Phomopsis subordinaria the impact of a decrease in infection probability on population growth rate is higher in a population on a rich soil than on a poor soil. This suggests that selection for increased resistance is more severe under the rich conditions. The second example concerns the demography of killer whale (Orcinus orca) pods. Although post-reproductive females, by definition, have no direct impact on the population growth rate, the analysis suggests that a higher probability of becoming post-reproductive may contribute positively to the rate of increase, because of its positive correlation with the fecundity of reproductive females in the pod.