This instalment sets up the question.The next instalment will answer it.
How I learned to stop worrying about chaos and love the Gompertz
INTRODUCTION
Robert May published a manuscript in 1974 showing that a simple deterministic density-dependent population model – the logistic equation - could exhibit complex dynamics. More specifically, chaotic dynamics (May, 1974). For many ecologists, the take-home message of Robert May’s work was that medium and long-term natural population fluctuations may be very difficult if not impossible to predict because natural population dynamics, like the weather, may often be chaotic. And chaotic dynamics are very sensitive to initial conditions and thus, tiny errors in initial population estimates would lead to large prediction errors. Interest in chaotic dynamics has remained high in five decades since the May paper was published though an analysis of publication rates suggests that interest in chaos in ecology peaked in 1996 (Munch et al 2022).
The debate about the ubiquity of chaotic dynamics in natural systems began shortly after the publication of the May paper and continues to today (Beddington et al. 1975, Gilpin 1979, Hastings 1993, Munch et al. 2022, Rogers et al. 2022). May and Oster (1976) made the explicit argument that ecologists should not be quick to attribute lack of predictability of population fluctuations to measurement, sampling or model error. That, in fact, natural populations may be commonly exhibiting chaotic dynamics and be subject to extreme sensitivity to initial conditions.
“Nevertheless, there is still a tendency on the part of most ecologists to interpret apparently erratic data as either stochastic “noise” or random experimental error. There is, however, a third alternative, namely, that wide classes of deterministic models can give rise to apparently chaotic dynamical behavior.”
Hastings et al (1993) made an enthusiastic case that chaos has important and substantive concerns for ecologists.
“We strongly believe that the study of chaos will yield important insights for ecologists. “
“We argue that chaotic dynamics are likely to be common rather than the exception in ecological systems by looking for chaos in ecological models, focusing only on biologically reasonable interactions and parameter values.”
For the first several decades, meta-analyses of natural population dynamics found little evidence that chaotic dynamics are common in nature (Turchin and Taylor, 1992, Ellner and Turchin 1995, Jaggi and Joshi 2001). More recently, Munch et al 2022, and Rogers et al have made the case that chaotic dynamics are common in nature. Both meta-analyses concluded that chaotic dynamics could be found in up to 30% of populations if (1) the underlying population model is multi-species rather than a one-dimensional discrete time model, (2) more sensitive detection methods, and (3) longer time series are used. However, the concerns raised by those early papers were that simple deterministic density-dependent models could lead to chaotic dynamics. I suspect that few ecologists would have been surprised that populations controlled by many drivers interacting in nonlinear ways could lead to complex dynamics that would be indistinguishable from chaos. What was surprising was that the ‘default’ population dynamics model that is routinely taught in introductory ecology courses could have chaotic dynamics.
One example of how embedded the concept of chaotic dynamics in scientific thought is that Cohen (1995) even referenced May’s work when discussing human carrying capacity despite no evidence that the logistic model captured human population dynamics better than other models and clear understanding that the intrinsic growth rather of human populations almost certainly didn’t fall above the r-thresholds of the logistic model.
Logistic population models were what May simulated in his early papers and are routinely taught as foundational ecological models in introductory ecology courses, but recently, Gebreyohannes and Houlahan (2024) have shown that another simple model – a Gompertz model – makes better predictions for natural populations than the logistic model. However, the underlying dynamics and the effects of r and K have not been explored in the literature.
The objectives of this paper are to (1) examine the simulated population dynamics of three simple population models - the differential logistic, the difference logistic and a Gompertz model – with particular emphasis on chaotic dynamics and (2) make recommendations about future population research.
