Environmental Versus Demographic Variability in Two-Species Predator-Prey Models

Abstract

We investigate the competing effects and relative importance of intrinsic demographic and environmental variability on the evolutionary dynamics of a stochastic two-species Lotka-Volterra model by means of Monte Carlo simulations on a two-dimensional lattice. Individuals are assigned inheritable predation efficiencies; quenched randomness in the spatially varying reaction rates serves as environmental noise. We find that environmental variability enhances the population densities of both predators and prey while demographic variability leads to essentially neutral optimization.PACS numbers: 87.23. Cc, 87.18.Tt The mathematical modeling of species interactions continues to be a central issue in population ecology [1][2][3][4]. Several simple models have been proposed, investigated, and sometimes realized under laboratory conditions. Yet more realistic and thus biologically more relevant model variants obviously have to include both external spatial disorder in the reaction rates to account for varying environmental conditions and intrinsic demographic heterogeneity stemming from trait variability in individuals. While we addressed the former in a recent study [5], our goal in this letter is to investigate the interplay between quenched spatial rate disorder and additional variability of individuals' reaction rates, as well as intriguing evolutionary co-optimization within interacting populations.We focus on the Lotka-Volterra (LV) predator-prey model owing to its simplicity and because its basic features are well-understood. It was first introduced to study fish populations in the Adriatic sea and chemical oscillations [6,7]. While the original deterministic LV (mean-field) equations yield neutral cycles and hence persistent nonlinear oscillations around a marginal fixed point [1], in stochastic implementations this species coexistence fixed point becomes stable and is approached very slowly through damped oscillations [8][9][10][11][12][13][14]. Spatially extended stochastic versions of the LV model yield striking dynamical patterns and emergent inter-species correlations [15][16][17][18][19][20][21] which may be utilized to quantitatively assess the response to external or internal changes. Population stability can be measured via the extinction time in small systems, where the stochastic kinetics ultimately reaches an absorbing zero-particle state [20,22].In our study of the effects of environmental rate variability in the LV model, we found a remarkable increase of the asymptotic population densities of both species with enhanced quenched spatial disorder, i.e., predation rates that are fixed to different lattice sites [5]. Yet the observed erratic population oscillations and relaxation towards the (quasi-)steady state occur on the time scale of many generations; for real biological systems, one therefore needs to address Darwinian evolutionary adaptation of individuals' traits. Consequently, we introduce fundamentally novel features by endowing individual predator and prey particles with randomly selected...

Eco-evolutionary dynamics of a population with randomly switching carrying capacity

“…To determine the fixation probability, in Ref. [39] we devised an effective approach, based on suitably choosing the parameter q (0 ≤ q ≤ b) and setting g(x) ≡ 1 + q in equation (8). This decouples N and x in an effective population whose size distribution, at quasistationarity and for any ν, is well described by the PDF (S3).…”

The effect of habitats and fitness on species coexistence in systems with cyclic dominance

“…In order to make our particle-based description more realistic, we introduce individual fitness and evolutionary adaptation by endowing every individual with a number f that reflects the varying efficiency of their predation and escape capabilities. Our setup follows general ideas introduced in (Dobramysl and Täuber, 2013;Chen et al, 2018) for predator-prey systems with one or two predators, but differs in the details of the implementation. This efficiency is assigned to an individual at its birth (or at the beginning of the simulation in case the individual is present when starting the run).…”

Evolutionary dynamics and competition stabilize three-species predator–prey communities

“…This does not reflect a natural environment where organisms from the same species may still vary in predation efficiency and death or reproduction rates because of their size, strength, age, affliction with disease, etc. In order to describe individually varying efficacies, we introduce a new character η ∈ [0, 1], which plays the role of an effective trait that encapsulates the effects of phenotypic changes and behavior on the predation / evasion capabilities, assigned to each individual particle [31]. When a predator A i (or B j ) and a prey C k occupy neighboring lattice sites, we set the probability (η Ai + η Ck )/2 [or (η Bj + η Ck )/2] for C k to be replaced by an offspring predator A z (or B z ).…”

“…Afterwards the remaining two species form a standard LV system and approach stable steady-state densities. Next we further modify the model by introducing evolutionary adaptation [31]. We also add a positive lower bound to the predator death rates in order to avoid 'immortal' particles.…”