A mathematical model to describe the demographic dynamics of long-lived raptor species

Highlights

• Problems on population dynamics of interest in biology and ecology are investigated.

• A mathematical model to describe the evolution of long-lived raptor species is introduced.

• Estimates for the reproductive parameters involved in the model are provided.

• An algorithm based on approximate Bayesian computation methods is proposed.

• The dynamics of the black vulture colony located at National Park of Monfragüe (Spain) is described.

Abstract

Population viability analysis of threatened large and long-lived raptor species has strong limitations due to the restricted demographic information available for these species. In this work, we mathematically model the demographic dynamics of these raptor species through time-indexed branching processes. By assuming the more general non-parametric statistical setting, we determine accurate estimates for the most relevant reproductive parameters involved in the model. To this end, we propose an algorithm based on approximate Bayesian computations methods. As illustration, by using real data of counts of the number of pairs in the population, we apply the proposed statistical and computational methods to describe the demographic dynamics of the Eurasian black vulture colony located at National Park of Monfragüe (Spain), which appears to be both the largest and densest breeding colony worldwide. In the scenario of these data-poor species, the class of time-indexed branching processes introduced appears to be appropriate and a more cost-effective method to evaluate dynamics and viability of the populations, applicable to the conservation of these taxa.

Introduction

Branching processes are appropriate mathematical models to describe the evolution of dynamical systems whose components, after certain life period, reproduce and die in such a way that the transition from one to other state of the system is made according to a certain probability distribution, for background see e.g. the monographies (Athreya and Ney, 2004, Asmussen and Hering, 1983, Guttorp, 1991). This kind of processes have been especially developed to describe biological phenomena, playing a major role in studies on population dynamics of great interest in conservation biology and population ecology, see e.g. Haccou et al. (2005) and Jagers (1975).

Significant efforts have been made to develop branching processes in discrete-time based on the assumption that the population dynamics of the species under consideration is properly described through generation-indexed processes, see Bruss and Slatvchova-Bojkova (1999), Corbacho et al. (2013), Molina et al., 2015, Molina et al., 2017, Mota et al. (2007), or Slatvchova-Bojkova (2000). This modeling assumes a non-overlapping situation about the individuals of the species respect to generations, namely, it is not possible the coexistence in the population of individuals from different generations. The non-overlapping assumption is not realistic to describe the population dynamics of many biological species. A more logical methodology is to model the fluctuations of such species through time-indexed instead of generation-indexed processes. In an epidemic framework, a time-indexed discrete-time process has been introduced in Penisson and Jacob (2012), however, concerning biological species, models based on time-indexed discrete-time branching processes have not been developed until now.

In this paper, we focus the attention on the study of mathematical models to describe the demographic dynamics of large and long-lived raptor species. The methodology based on population viability analysis (PVA), considered in conservation biology and in the management of threatened or endangered species, usually requires information about several variables (sizes, ages, mortality rates, growth rates, environmental variables, etc.), see Margalida et al. (2015) and Tauler et al. (2015). In practice, taking into account the characteristics of such raptor species, real data about such variables are very difficult to obtain. Consequently, PVA is not usually feasible to apply, see Margalida et al. (2015). Mathematical models based on others methodologies have not been sufficiently investigated in the literature on the population dynamics of these species.

The motivation behind the present work, is to develop a new class of time-indexed discrete-time processes, to describe the demographic dynamics of large and long-lived raptor species, which requires an information feasible to be observed. By considering such a class of models, we will investigate several inferential problems of ecological interest for these species. In particular, with the aim to forecast possible changes in their population dynamics, it is of great practical importance to determine close estimates for the main parameters controlling their reproduction, namely, the reproduction success probabilities (rates) for a female at different ages during her fertile life. It is important to point out that by success we mean not only the survival of the chicks but also their future integration into the breeding population of their natal area. By using, as mathematical tool, the so-called approximate Bayesian computation methodology, we will determine accurate estimates for such reproductive parameters. Then, from the adjusted model, we will get a reasonable description on the demographic dynamics of these raptor species.

The work is organized as follows. In Section 2, the mathematical model is formally described and intuitively interpreted. Section 3 is devoted to determining, under the more general non-parametric statistical setting, estimates for the success probabilities of a female at different ages during her fertile life. To this end, an algorithm based on approximate Bayesian computation methods is proposed and commented. In Section 4, by using real data of counts of the number of pairs in the population, we apply the proposed methodology to describe the demographic dynamics of the black vulture colony located at National Park of Monfragüe (Spain). With the purpose of assessing the accuracy of the estimates for the parameters, a sensitivity analysis about the main elements considered in the practical application of the algorithm, especially metric and tolerance level, is performed. The concluding remarks and some questions for research are included in Section 5.