Revisiting the logistic map: A closer look at the dynamics of a classic chaotic population model with ecologically realistic spatial structure and dispersal
Introduction
Although the chaotic systems literature has grown over the past several decades, the bulk of it resides in the physical sciences (e.g. fluid dynamics) and is neither written for, nor read by ecologists. Early conceptual papers introduced chaotic concepts to the ecological literature (e.g., May, 1973; May, 1976; Schaffer, 1985; Hastings, 1993; Hastings et al., 1993) but computational limitations at the time restricted such research to simple models that are difficult to apply to real populations. Here, we revisit and build upon these earlier models by performing a closer examination of a chaotic population within an ecologically applicable framework. We create a chaotic growth-dispersal model with an explicit 1-dimensional spatial component, and detail dynamics for a large parameter space, examine the fractal-like boundaries between differing population dynamics regimes, and add stochastic elements to simulate spatio-temporal heterogeneity and observe the interactions between chaotic dynamics and stochasticity. In this way we add to the ecologists’ chaotic toolbox by detailing the spectrum of behaviors possible for such a system. Incorporating enough ecologically realistic components insures that the model dynamics can be compared to the dynamics of real populations.
Chaos has been detected in many and diverse biological systems, such as the spread of infectious disease (Sugihara and May, 1990), marine environments (Glaser et al., 2014), and plant growth dynamics (Billings et al., 2015). These studies focus on detection of chaos as a way to understand system stability and future behavior. There are few examples of directly modeled nonlinear populations, but Cushing et al. (2003) provide a thorough analysis of the nonlinear dynamics of the flour beetle, with both a nonlinear population model and actual experimental data to verify the model. We envision our model as a stepping stone for further direct applications of nonlinear models to real populations, as our generalized model provides a mechanism for deeper understanding of the full range of dynamics present in a chaotic population, as well as a means of understanding the biological cause of behavioral changes in a population as ecologically relevant parameters change.
For example, harvesting of cod affects both the growth rate and dispersal of the population along its entire range. For cod in the Gulf of Maine, surplus production models illustrate how growth rates first increase (due to density dependence) to compensate for increased harvesting, and then decrease if the population is overfished. In heavily depleted, and therefore likely chaotic conditions (Glaser et al., 2014), the effects of this variability on cod dispersal (and, thus, availability to the fishery) are poorly understood and can result in management failures (NEFSC 55th SAW, 2013). For this reason, a better method for understanding changes in movement and dispersal in chaotic populations is essential to sustainable management.
The earliest introduction of chaotic population models to the ecological literature focused on zero-dimensional (zero spatial dimension, one-dimensional phase space) single-species models (May, 1973, May, 1976). Subsequent models added spatial dimension in the form of discrete patches, where a certain percentage of the population in one patch disperses to the adjacent patch(es) with each iteration of the model (Hastings, 1993; Kaneko, 1984; Willeboordse, 2003; White and White, 2005). When enough patches are incorporated, the model simulates a growth and diffusive dispersal system. However, such models employ numerical diffusion as a dispersal mechanism, which permits limited diffusion per iteration/generation, as the coupling strength parameter must remain small for the validity of the system (Kaneko, 1989). A subset of the literature expands the diffusive range by exploring globally coupled patches (Kaneko, 1992, Solé et al., 1992, Willeboordse, 2003, Willeboordse, 2002), but these are limited to an even smaller portion of the population diffusing into neighboring patches, and each patch receives a uniform proportion of the population regardless of its proximity to the parent patch. Additionally, such models largely employ periodic boundary conditions, an ecologically unrealistic feature (Aiken and Navarrete, 2014).
Improvements upon the numerical diffusion models were made when the diffusive dispersal mechanisms were replaced with dispersal kernels (e.g. Ruxton and Doebeli, 1996; Doebeli and Ruxton, 1998; Saravia et al., 2000; Labra et al., 2003). The kernels provide a more realistic dispersal scenario than discretized diffusion, as the dispersal distance is easily changed, and neighboring discrete locations generally receive a larger percentage of neighboring populations than far away locations.
Many of the aforementioned models share a common model of growth: the logistic map. The logistic map represents one of the simplest difference equations with chaotic dynamics, popularized by Robert May (1976) as a discretization of the logistic equation: where is the growth parameter, is the discrete time index, and is a dimensionless population (in May’s model, for outside of the range solutions will diverge and therefore lose physical meaning). There is no spatial structure in this model. This “classic logistic map” can represent an organism that reproduces in discrete generations with no overlap (e.g., annual plants, some anadromous fish, some benthic invertebrates, and many insect populations). An array of complicated population dynamics result from this equation, depending on the growth parameter. For the population reaches steady state after the initial growth period. As increases, the population experiences a period doubling bifurcation cascade until eventually descending into chaos. Doubling begins for and the onset of chaos begins for (May, 1976). Beyond the beginning of chaos, the logistic map has pockets of periodicity at special values of , e.g., exhibits a period-3 population cycle (Strogatz, 1994). The equation has some ecologically impractical constraints, such as population extinction for and (May, 1976).
A family of similarly simplistic models exhibit great dynamical complexity (see May and Oster, 1976), and the Ricker model displays qualitatively indistinguishable dynamics to the logistic map (e.g., Andersen, 1991). Indeed, the bifurcation diagrams for the two models look incredibly similar. Besides displaying chaotic dynamics over a particular parameter range, these models exhibit other interesting dynamical behaviors, such as long periods of transience (e.g. Hastings and Higgins, 1994; Saravia et al., 2000; Labra et al., 2003).
Our model starts with the classic logistic map as one of the simplest models of growth. We employ a normalized Gaussian kernel as the dispersal mechanism, which is a more flexible and ecologically applicable dispersal framework than discretized diffusion (and dispersal kernels have a long and established history of use in the ecological modeling literature, e.g., Chesson and Lee, 2005). The dispersal kernel has several advantages, such as decoupling the scale and length of dispersal distance from the spacing of discrete patches, as well as providing a mechanism to easily change the dispersal distance.
Gaussian kernels are found in seed dispersal models (e.g., Clark et al., 1999), planktonic dispersal models (e.g., Byers and Pringle, 2006), and myriad other ecological models. Generally, diffusive-type dispersal effectively models the random dispersal of offspring away from parents. As the number of random movements away from the parent increases, the shape of these random movements forms a Gaussian. This is called the “random walk” model, and is appropriate for small animals such as insects (Skellam, 1951). Diffusive-type spread also applies to animals moving over favorable terrain over multiple generations (Krebs, 2009, Pielou, 1979), or immobile species transported by wind or water (Okubo, 1980).
Although diffusive-type dispersal is a useful simplification, there exist many alternative dispersal mechanisms, e.g., non-Gaussian kernels (Kot et al., 1996, Chesson and Lee, 2005, Pringle et al., 2009) and asymmetrical advective dispersal (Okubo, 1980, Byers and Pringle, 2006, Lutscher et al., 2010); these will be included in our future work. However, for this paper, we start with the simplest case and limit ourselves to the symmetrical dispersal-only model. Additionally, we chose to work in a one-dimensional domain, which is an applicable simplification for certain environments, including edge communities bounded by two distinct habitats such as riparian, littoral or coastal areas.
We test ecologically relevant absorbing (dissipative) boundary conditions, and these boundary conditions serve as the basis for the bulk of our ecological analysis. We additionally test periodic boundary conditions to provide deeper insight and as an illustrative comparison with previous literature and with the classic logistic map. However, these boundaries are not ecologically useful in most real-world systems, as opposite ends of a domain are neither correlated nor connected. Furthermore, periodic boundaries cannot serve as a proxy for an infinite domain. An infinite domain naturally explores the limit of small dispersal (with respect to domain size). Because our dispersal distance expresses length relative to domain size, larger dispersal distances would effectively model the biologically implausible case of infinite dispersal. Boundaries in which 5% of the population survives over the boundary edge and reflecting boundaries were tested to compare with earlier work (because they largely resemble the absorbing and periodic boundary cases, respectively, results are presented in the supplementary material; see Appendix A).
With our relatively simple model we partake in a closer examination of the complex dynamics present in a chaotic population while utilizing ecologically reasonable dispersal and boundary conditions. We illustrate how minute changes in dispersal distance, habitat size, or growth rate can dramatically alter the qualitative dynamics of a population (e.g. periodic behavior shifting to chaotic, or vice-versa), and how the boundaries between qualitatively different behaviors may also be complicated. We additionally test a population’s response to stochastic elements, simulating the spatial heterogeneity found in real-world populations. Understanding the dramatic and abrupt behavioral changes possible in chaotic populations has significant ecological applications.
Section snippets
Methods
We explore the logistic map applied to a spatially discrete population along a one-dimensional domain. This is a density-dependent model, where larval production in a discrete patch at time is dependent on the population of that patch at time . The domain is homogeneous. The number of propagules produced is given by a modified logistic map:
This is the “generalized logistic equation” (Li and Yorke, 1975). All variables are dimensionless. Lowercase and are
Results
Fig. 1 illustrates the chaotic parameter space of the classic logistic map for growth rates to , calculated using the norm of differences method above. Red indicates chaotic population behavior while grey indicates non-chaotic behavior.
Results are presented for absorbing boundary conditions unless otherwise stated. Fig. 2 provides an example of the four categories of population distribution patterns witnessed. These patterns are referred to as the (a) pattern with kinks (b) frozen
Model dynamics
With our simple model, we witness a variety of behaviors, dependent on the growth rate and dispersal distance/domain size. This system illustrates how well-behaved periodic population dynamics can actually be driven by equations which retain the potential for chaotic dynamics, given a different parameter set or an external perturbation. Change in growth rate, dispersal distance, or domain size could shift the population into a different behavioral regime. Such dramatic change in behavior would
Conclusion
Classic models from the 1970s through 1990s introduced chaotic theory into the ecological literature. The applications for these models were largely theoretical, given the simplicity of the models and the computational limitations of the time. Here, we revisit these classic models with modern numerical techniques to examine the systems more closely and to add elements of ecological realism such that the models can be employed for concrete ecological applications. Combining improved knowledge of
Acknowledgment
Funding for L.S. Storch and J.M. Pringle was provided by the United States National Science Foundation (NSF) award OCE 0961344.
Conflict of interest: The authors declare that they have no conflict of interest.
References (44)
Properties of some density-dependent integrodifference equation population models
Math. Biosci.
(1991)- et al.
Families of discrete kernels for modeling dispersal
Theor. Popul. Biol.
(2005) Pattern dynamics in spatiotemporal chaos: Pattern selection, diffusion of defect and pattern competition intermettency
Physica D
(1989)Spatiotemporal chaos in one- and two-dimensional coupled map lattices
Physica D
(1989)- et al.
Spatial structure, environment heterogeneity, and population dynamics: Analysis of the coupled logistic map
Theor. Popul. Biol.
(1998) - et al.
Relating coupled map lattices to integro-difference equations: dispersal-driven instabilities in coupled map lattices
J. Theor. Biol.
(2005) - et al.
Coexistence of competitors in marine metacommunities: Environmental variability, edge effects, and the dispersal niche
Ecology
(2014) - et al.
Nonlinear tree growth dynamics predict resilience to disturbance
Ecosphere
(2015) - et al.
Going against the flow: Retention, range limits and invasions in advective environments
Mar. Ecol. Prog. Ser.
(2006) - et al.
Studying edge geometry in transiently turbulent shear flows
J. Fluid Mech.
(2014)
Seed dispersal near and far: Patterns across temperate and troical forests
Ecology
Chaos in Ecology: Experimental Nonlinear Dynamics
Stabilization through spatial pattern formation in metapopulations with long-range dispersal
Proc. R. Soc. Lond. [Biol.]
Complex dynamics may limit prediction in marine fisheries
Fish Fish.
Complex interactions between dispersal and dynamics: lessons from coupled logistic equations
Ecology
Persistence of transients in spatially structured ecological models
Science
Chaos in ecology: Is mother nature a strange attractor?
Annu. Rev. Ecol. Syst.
Period-doubling of kink-antikink patterns, quasiperiodicity in antiferro-like structures and spatial intermittency in coupled logistic lattice. Towards a prelude of a “field theory of chaos”
Progr. Theoret. Phys.
Overview of coupled map lattices
Chaos
Disperal data and the spread of invading organism
Ecology
Population dynamics of large and small mammals: Graeme Caughley’s grand vision
Wildl. Res.
Dispersal and transiet dynamics in metapopulations
Ecol. Lett.
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