Abstract
The notion of a potential-growth indicator came to being in the field of matrix population models long ago, almost simultaneously with the pioneering Leslie model for age-structured population dynamics, although the term has been given and the theory developed only in recent years. The indicator represents an explicit function, R(L), of matrix L elements and indicates the position of the spectral radius of L relative to 1 on the real axis, thus signifying the population growth, or decline, or stabilization. Some indicators turned out to be useful in theoretical layouts and practical applications prior to calculating the spectral radius itself. The most senior (1994) and popular indicator, R0(L), is known as the net reproductive rate, and we consider two others, R1(L) and RRT(A), developed later on. All the three are different in terms of their simplicity and the level of generality, and we illustrate them with a case study of Calamagrostis epigeios, a long-rhizome perennial weed actively colonizing open spaces in the temperate zone. While the R0(L) and R1(L) fail, respectively, because of complexity and insufficient generality, the RRT(L) does succeed, justifying the merit of indication.
Highlights
The concept of the potential-growth indicator (PGI) was developed in the theory of matrix population models (MPMs) for the dynamics of discrete-structured biological populations [1,2]
The pattern of matrix L corresponds to the associated directed graph [3], which is called the life cycle graph [1] (LCG) as it represents graphically the biological knowledge of life histories involved into the model and the way the population structure is observed in the field or laboratory (Figure 1 gives an example)
The history of PGIs in matrix population models, which began with the simplest expression for the Leslie matrix three quarters of a century ago [32], has come to its logical end with the explicit formulas for R0(L), R1(L), and RRT(L), where L is a population projection matrix (PPM) representable as L = T + F (2)
Summary
The concept of the potential-growth indicator (PGI) was developed in the theory of matrix population models (MPMs) for the dynamics of discrete-structured biological populations [1,2]. This kind of model represents the basic tool in the mathematical demography of plant and animal populations structured with regard to a certain classification trait such as the age, size, or developmental stage of individuals in a local population of a given species [1,2]. The LCG may be strongly connected [3], signifying a certain integrity of the individuals’ life history and providing for the PPM being irreducible [4,5] (or indecomposable in the other terminology [6]), but it ceases to be strong when including post-reproductive stages (further examples follow)
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