1 Introduction

Predicting the risk of population extinction is one of the central themes in population biology (Lande et al., 2003; Ovaskainen and Meerson, 2010; Ladle, 2009). Population extinction is influenced by many unpredictable factors, such as the environmental conditions (e.g., food/water availability, climate change) and demographic variability (e.g., genetic diversity, different fitness of individual species); see Bartlett (1961, 2000), McLaughlin et al. (2002), and Lande and Orzack (1988). Understanding and predicting the time to extinction is important in the context of the conservation of biodiversity, control of epidemics and for planning a responsible consumption of natural resources (Davidson et al., 2009; Brook et al., 2000). Extinction studies are also relevant in a broader context of complex systems exhibiting multiagent interacting dynamics: stock market trading (Sprott, 2004), biochemical reactions (Wilkinson, 2013), brain activity networks (Rabinovich et al., 2008), and particle systems with complex pair interaction (Krapivsky et al., 2010).

The events of species extinction and survival are inherently stochastic. Stochasticity becomes the dominant factor of system evolution, particularly when the number of species becomes very low. This implies that when the fluctuating population size of the system falls below a certain threshold, then the demographic noise (i.e., noise induced by finite population size) drives the system to extinction (Ovaskainen and Meerson, 2010; Dennis et al., 1991).

The theoretical prediction of a population extinction event, therefore, involves the theory of stochastic processes, Ovaskainen and Meerson (2010), Dennis et al. (1991). More specifically, the population dynamics are modelled as a stochastic birth-and-death process and the theory predicts the evolution of the probability distribution function of the system state in a multidimensional state space (Kolmogorov or Fokker-Plank equations). The event of population extinction occurs when the system trajectory crosses the surface of a given population threshold in the state space. The extinction time then simply corresponds to the first-passage time of the underlying random process. A plethora of theoretical methods exists for studying the first-passage time problems; see Gardiner (2010) and Redner (2001). Depending on the complexity of the underlying birth-and-death process, the first-passage time approach may lead to closed-from analytical expressions, which relate the extinction time and the parameters of the model. These expressions can be used as a foundation for development of an algorithm for prediction of population extinction, based on the temporal observation of the species count, followed by the estimation of system model parameters.

For the case of no interaction among the species (i.e., a single species in isolation), it has been found that the population size undergoes a random walk, leading to the inverse Gaussian distribution of extinction times; see Lande and Orzack (1988) and Dennis et al. (1991). For the case of structured populations consisting of multiple interactive groups, the estimation of extinction time becomes much more challenging. This is due to the temporal correlations between population groups, which drives a complex and versatile set of extinction scenarios (Ovaskainen and Meerson, 2010). Close to extinction, the system exhibits strong fluctuations and becomes analytically intractable. The conventional approach to overcome this difficulty is to invoke simplifying assumptions (e.g., moments closure scheme or mean-field theories (Allen, 2003; Holyoak et al., 2000). While these assumptions lead to a tractable problem, they often have limited validity—for comments on inconsistency of the moment closure approach, see Gardiner (2010); or impose a significant constraint on model parameters—e.g.,, diffusion approximation is valid only for large population sizes (Gardiner, 2010). A novel approach to this problem was reported in recent studies, which demonstrated that the longtime phenomenology of population extinction can be described by methods borrowed from quantum mechanics (Goldenfeld, 1984; Dykman et al., 1994; Doering et al., 2005). These methods recently helped to understand the extinction of dynamic systems consisting of two interacting species [e.g., SIS compartmental model of an epidemic outbreak (Ovaskainen and Meerson, 2010) and predator-prey model (Parker and Kamenev, 2009, 2010).

In this chapter, we go a step further and study the extinction of a complex biological system consisting of many species that compete for a finite set of resources. The problem is cast in the framework of a stochastic multiple-predator single-prey Lotka-Volterra (LV) system (Rabinovich et al., 2008). In this context, the prey population plays the role of a finite supply of food or resources, for which the predators compete. Extinction is the event when either the resources are exhausted or all the competing predators die out.

The theoretical predictions of extinction time build upon the previous work of Parker and Kamenev (2009, 2010) for the classical LV system (one predator and one prey). We extend these studies by introducing a concept of an aggregated predator, whose effect in the multipredator system is approximately equivalent to the combined effect of all predators. This coarse approximation of the multipredator LV system removes all the irrelevant information on predators (including their total number and the specifics of each group) resulting in a significant simplification of estimation algorithms (the reduced state space and independence of the variety of predator-species).

The chapter also develops a practical algorithm for forecasting extinction. Given noisy and sporadic observations of the prey count (the quantity of resources), it is capable of predicting the timing of extinction event in a probabilistic manner. Remarkably, in doing so, it does not have to know how many predators are competing for resources. The first phase of this algorithm is parameter estimation: observations of the quantity of finite resources are used to estimate the unknown system parameters via a Bayesian method known as particle Markov Chain Monte Carlo (pMCMC) (Andrieu et al., 2010). These estimated parameters, along with the analytical expressions derived here, are then used in the second phase of the algorithm to compute the probability density function (PDF) of extinction time. Validation of the resulting, theoretically predicted, PDF of extinction time is carried out numerically against the actual extinction statistics.

2 A Model of Competing Species

As a model of competing species, we adopt the multipredator, single-prey LV system. Let x be the prey population that acts as the food for n competing predators y_i, $i=1,\cdots ,n$. Assuming that the population sizes are continuous-valued and the system evolution is deterministic, it can be described by $n+1$ ordinary differential equations (Korobeinikov, 1999):

$\dot{x}=\alpha x\left(1-\sum _{i=1}^n{\beta }_i{\gamma }_i\right)$

$y_i={\beta }_{i}x{y}_i-{\gamma }_i{y}_i$

for $\alpha ,{\beta }_i,{\gamma }_i>0$ and $i=1,\cdots ,n$. In this model, the prey species reproduces at the rate α, while the predator species i die (by natural causes) at rate γ_i. By consuming the prey, predator i reproduces at the rate β_i. Note that according to this model, predator species do not interact directly; i.e., they do not target each other. Each predator is thus characterized by the survival capacity (fitness) β_i/γ_i. Korobeinikov (1999) showed that an unsuccessful predator, with the smallest survival capacity, will be extinct in finite time. However, the overall LV system [Eqs. (25.1)–(25.2)] is stable, providing that all survival capacities are different. The system [Eqs. (25.1)–(25.2)], with the state space $\left(x,y_1,\cdots ,y_n\right)$, has $n+1$ equilibria: the trivial one at the origin, and n solutions in coordinate planes x − y_i, given by: $\left(\frac{{\gamma }_1}{{\beta }_1},\frac{\alpha }{{\beta }_1},0,\cdots ,0\right)$, $\left(\frac{{\gamma }_2}{{\beta }_2},0,\frac{\alpha }{{\beta }_2},0,\cdots ,0\right),$ $\cdots$, $\left(\frac{{\gamma }_n}{{\beta }_n},0,\cdots ,0,\frac{\alpha }{{\beta }_n}\right)$.

An extinction event (in this case, when either the prey becomes extinct or all the competing predators die out) can happen only if the system is modelled in a discrete-stochastic manner (Ullah and Wolkenhauer, 2011). For this purpose, let us represent the single-prey, multipredator LV system (abbreviated to LV-1n) by a system of $2n+1$ biochemical reactions:

$X\stackrel{\alpha }{\to }2X$

$X+Y_i\stackrel{{\beta }_i}{\to }2Y_i$

$Y_i\stackrel{{\gamma }_i}{\to }0$

for $i=1,\cdots ,n$. Stochastic fluctuations (demographic noise) in the LV-1n system [Eqs. (25.3)–(25.5)] will inevitably force it to extinction, regardless of the initial count of species and the values of rate parameters $\alpha ,{\beta }_1,\cdots ,{\beta }_n,{\gamma }_1,\cdots ,{\gamma }_n$.

Figure 25.1 shows two trajectories of the LV-1n system for n = 2 in the (x, y₁, y₂) space, using parameters α = 15, β₁ = 0.02, β₂ = 0.015, γ₁ = 5 and γ₂ = 4 with initial conditions x(0) = 150, y₁(0) = 225 and y₂(0) = 112. The plot in Figure 25.1(a) is generated using the continuous-deterministic model [Eqs. (25.1)–(25.2)]; note that the population of the second predator gradually dies out due to its lesser survival capacity; the system reaches an equilibrium in the form of a closed orbit in the (x, y₁, 0) plane, describing the stable periodic oscillations of the first predator and the prey. The trajectory in Figure 25.1(b) is created using the discrete-stochastic model [Eqs. (25.3)–(25.5)], implemented using the Gillespie algorithm (Gillespie, 1977; Ullah and Wolkenhauer, 2011). Note that extinction happens when the trajectory touches one of the axes of the state space (x, y₁, y₂).

3 Density function of extinction time

In order to derive the PDF of extinction time for the LV-1n system, first we conceptually reduce the original model given by Eqs. (25.1)–(25.2) in the continuous-deterministic form or [Eqs. (25.3)–(25.5)] in the discrete-stochastic form, by introducing an aggregated predator y. Then the original LV-1n model [Eqs. (25.1)–(25.2)] can be mapped to the classical LV-11 model with one prey and one predator (i.e., with n = 1) as follows:

$\dot{x}=\alpha x\left(1-\beta y\right)$

$y=\beta xy-\gamma y\text{.}$

Similarly, Eqs. (25.3)–(25.5) can be represented by three reactions:

$X\stackrel{\alpha }{\to }2X,X+Y\stackrel{\beta }{\to }2Y,Y\stackrel{\gamma }{\to }0\text{.}$

Note that we have introduced the effective parameters β and γ of the aggregated predator y, which are related to the (unknown) original parameters ${\beta }_i,{\gamma }_i,i=1,\cdots ,n\text{.}$

For the classic LV-11 model with a single predator [cf. Eq. (25.8)], analytical results for extinction time exist. According to Parker and Kamenev (2009, 2010), the PDF of the rescaled dimensionless extinction time $\tilde{T}=T/\tau$ can be presented in terms of a parameter-free inverse Gaussian distribution (note that the standard inverse Gaussian distribution, derived as the first-passage time of the Brownian motion with a drift, has two parameters):

$P\left(\tilde{T}\right)=\frac{a}{\sqrt{\pi {\tilde{T}}^3}}\mathit{exp}\left[-{\left(\tilde{T}-a\right)}^2/\tilde{T}\right],$

where $a\approx 0.5$ is a constant, found empirically and specific to the LV model [Eq. (25.8)]. The characteristic extinction time τ is a function of reaction rates α, β, γ and the initial state of the aggregated system x(0), y(0). It is given by

$\tau =\frac{1}{\beta }\frac{B\left(s\right)}{{ϵ}^2+1/{ϵ}^2},$

where $ϵ=\sqrt{\alpha /\gamma }$ is the prey-predator asymmetry parameter and the dimensionless function B(s) specifies the dependence on the initial condition $s=G_o/\sqrt{\alpha \gamma }$, with

$G_o=\beta y\left(0\right)-\alpha -\alpha ln\left(\frac{y\left(0\right)\beta }{\alpha }\right)+\beta x\left(0\right)-\gamma -\gamma ln\left(\frac{x\left(0\right)\beta }{\gamma }\right)\text{.}$

B(s) is approximately given by

$B\left(s\right)\approx \frac{1+Q{s}^2}{1+Qscosh\left(s\right)},$

with $Q\left(ϵ\right)=\left({ϵ}^2+\frac{1}{{ϵ}^2}\right)/\left(ϵ+\frac{1}{ϵ}\right)$.

In summary, we can theoretically predict the PDF of extinction time T of the population system with competing predator species if we can determine the characteristic time τ of the equivalent LV-11 model. Since τ is the known function of reaction rates α, β, γ and the initial conditions x(0) and y(0), our next task is to compute the estimates of these quantities [i.e., $\hat{\alpha },\hat{\beta },\hat{\gamma },\hat{x}\left(0\right),\hat{y}\left(0\right)$]. Estimation needs to be carried out using the noisy counts of prey species (food, resources) collected occasionally during an observation period.

4 Estimation of parameters

Let an observation (count) of prey species of the LV-1n system at time t_k be denoted as z_k. Observations are modeled as random draws from the Poisson distribution whose parameter is the true count of prey species at time t_k. Let us denote all observations collected during the observation period as a time series $z_{1:K}={\left\{z_k\right\}}_{1\le k\le K}$, with index k referring to the sampling time t_k. Note that prediction is carried after the observation period; that is, starting from time t_K, when the last observation z_K is reported. This means that the initial time for prediction in Eq. (25.11)—that is, $t=0$ —refers to time t_K. Hence, we are after the estimates $\hat{x}\left(t_K\right)$, $\hat{y}\left(t_K\right)$, in addition to $\hat{\alpha },\hat{\beta },\hat{\gamma }$.

Parameter estimation is carried out in the Bayesian framework assuming the reduced LV-11 model. Let $\mathbf{\theta }={\left[\begin{array}{ccc}\alpha & \beta & \gamma \end{array}\right]}^T$ denote the unknown parameter vector and ${\mathit{x}}_t={\left[\begin{array}{cc}x\left(t\right)& y\left(t\right)\end{array}\right]}^T$ the (unknown) state vector of the LV-11 system during the observation interval; i.e., $t_1\le t\le t_K$. The goal in the Bayesian framework is to compute the posterior distribution $p\left(\mathit{\theta },{\mathit{x}}_{t_1\le t\le t_K}\right)$, which provides the complete probabilistic description of all unknown quantities. From this posterior, one can extract the marginal distributions $p\left(\mathit{\theta }|z_{1:K}\right)$ and $p\left({\mathit{x}}_{t_K}|z_{1:K}\right)$, and subsequently point estimates $\hat{\alpha },\hat{\beta },\hat{\gamma }$, $\hat{x}\left(t_K\right)$, $\hat{y}\left(t_K\right)$, which are required for the computation of the characteristic time τ. Since the state of the LV system is time-varying and its dynamics is highly nonlinear, estimation will be carried out using the sequential Monte Carlo method. However, the straightforward sequential Monte Carlo estimation on the joint space $\left(\mathit{\theta },{\mathit{x}}_{t_1\le t\le t_K}\right)$, where part of the state vector are fixed parameters, is known to be inefficient due to “particle degeneracy” (Liu and West, 2001). Instead, it is more efficient to perform estimation separately on θ and ${\mathit{x}}_{t_1\le t\le t_K}|\mathit{\theta }$ spaces, using the pMCMC method (Andrieu et al., 2010).

For a given θ, estimation of the posterior p(x_t| z_1 : K, θ) in interval $t_1\le t\le t_K$ is carried out using the standard particle filter (PF) (Ristic et al., 2004; Cappé et al., 2007), developed for the stochastic dynamic model specified by reactions [Eq. (25.8)]. The main feature of this PF is that it uses the Gillespie algorithm (Gillespie, 1977; Ullah and Wolkenhauer, 2011) as an exact discrete-time stochastic simulation method to predict the transition of particles from t_k to $t_{k+1}$. In the update step of the PF, the Poisson distribution acts as the likelihood function of observations z_1 : K. The PF performs two roles. First, at each time t (where $t_1\le t\le t_K$), it provides a random sample $\left\{{\mathit{x}}_t^{\left(1\right)},\cdots ,{\mathit{x}}_t^{\left(N\right)}\right\}$, which approximates the posterior p(x_t| z_1 : K, θ); here, N is the number of random samples or particles (recall that only the last random sample, corresponding to the time t_K, is required for the estimation of τ). Second, the PF provides an estimate of the marginal likelihood $g\left(z_{1:K}|\mathit{\theta }\right)$ by integrating out the state x_t at each time step (Cappé et al., 2007). This marginal likelihood $g\left(z_{1:K}|\mathit{\theta }\right)$ is required in the MCMC Metropolis-Hastings (MH) scheme (Robert and Casella, 2004) for estimation of the posterior $p\left(\mathit{\theta }|z_{1:K}\right)$ according to the Bayes rule: $p\left(\mathit{\theta }|z_{1:K}\right)\propto g\left(z_{1:K}|\mathit{\theta }\right)\pi \left(\mathit{\theta }\right)\text{.}$ Here, π(θ) is the prior distribution of the parameter vector θ (reflects our prior knowledge of rate parameters). Thus, the PF plays a double role in the pMCMC method. Note that at every iteration of the MH algorithm, it is necessary to run the PF in order to obtain the estimate of $g\left(z_{1:K}|\mathit{\theta }\right)$. The output of the MH algorithm is a random sample of parameter vectors, ${\left\{{\mathit{\theta }}^{\left(j\right)}\right\}}_{1\le j\le M},$ which approximates the posterior $p\left(\mathit{\theta }|z_{1:K}\right)$; here, M is the number of samples generated by the MH algorithm.

5 Numerical results

A numerical study was carried out for the LV-1n system with n = 3 competing predators. Figure 25.2 shows the count of species for the considered LV-13 system during an observation period that lasts 3 units of time. Red squares indicate the observations of prey species collected every 0.25 units of time. The parameters of this LV-13 system are α = 5, β₁ = 0.15, β₂ = β₃ = 0.01, γ₁ = 4, and γ₂ = γ₃ = 3. The state vector at the final observation time $t_K=3$ is x_o = ${\left[\begin{array}{cc}\begin{array}{cc}x\left(t_K\right)& y_1\left(t_K\right)\end{array}& \begin{array}{cc}{y}_2\left(t_K\right)& y_3\left(t_K\right)\end{array}\end{array}\right]}^T$ = ${\left[\begin{array}{cc}\begin{array}{cc}73& 161\end{array}& \begin{array}{cc}120& 101\end{array}\end{array}\right]}^T$.

Figure 25.3 shows (a) the empirical PDF and (b) the corresponding cumulative distribution function (CDF) of extinction time (black solid lines). The PDF was obtained by the kernel density estimation (Silverman, 1986), based on 1000 samples of extinction time. The samples of extinction time were generated by running the LV-13 system with true parameters α = 5, β₁ = 0.15, β₂ = β₃ = 0.01, γ₁ = 4, γ₂ = γ₃ = 3 and identical initial state x(0), until either the prey count or the sum count of all predators equals zero.

Figure 25.3 also shows 100 overlaid theoretical PDFs and CDFs computed using 100 pMCMC-generated samples of estimates $\hat{\alpha },\hat{\beta },\hat{\gamma }$, $\hat{x}\left(0\right),\hat{y}\left(0\right)\text{.}$ The number of particles in the PF of the pMCMC was N = 1000. The MH algorithm of pMCMC was run M = 1000 times, but only the last 100 samples were used in plotting the theoretical PDFs of extinction time. We can observe from Figure 25.3 that there is a good match between the theoretically predicted PDFs/CDFs and the empirical PDF/CDF, which verifies the proposed method. The theoretically predicted PDFs/CDFs are computed without knowledge of the number of predator species competing for the prey, because all predator species are treated as one aggregated predator.

6 Summary

This chapter described an extinction process in a complex biological system with competition. To model this process, we used a generalized version of the stochastic LV system (multiple-predator and single-prey). Given noisy and sporadic observations of the prey count only, collected during an observation period, the proposed method is capable of predicting the timing of extinction in a probabilistic manner. Remarkably, in doing so, this method does not require prior knowledge of how many predators are competing for food (i.e., prey). The key idea is to treat the competing predator species as a single, aggregated predator, for which the theoretical PDF of extinction time has been derived using the mathematical tools of the first-passage-time studies.

The LV system is used as a canonical model in many areas of population biology and, by a well-known mapping procedure (Bomze, 1995), can be related to the replicator equation that is an important part of a modeling framework in evolutional studies (Hofbauer and Sigmund, 1998). From this perspective, the presented study can be seen as a step toward the goal of long-term prediction of abrupt changes (catastrophic events) in complex dynamic stochastic systems, operating under supply-demand constraints (e.g., emerging epidemic, virus mutation, stock-market crash, network traffic jam).