The logistic equation and a critique of the theory of natural selection

Species coexistence is one of the central themes in modern ecology. Coexistence is a prerequisite of biological diversity. However, the question arises how biodiversity can be reconciled with the statement of competition theory, which asserts that competing species cannot coexist. To solve this problem natural selection theory is rejected because it contradicts kinetic models of interacting populations. Biological evolution is presented as a process equivalent to a chemical reaction. The main point is that interactions occur between self-replicating units. Under these assumptions biodiversity is possible if and only if species are identical with respect to the patterns of energy flow in which individuals are involved.

💡 Research Summary

The paper tackles one of the most persistent puzzles in modern ecology: how can the extraordinary diversity of life be reconciled with the competitive exclusion principle, which states that two species that vie for the same limiting resource cannot coexist indefinitely? The authors begin by reviewing the classical competition theory and the standard formulation of natural selection. They argue that the two frameworks, when taken together, generate a logical inconsistency. Competition theory predicts that one species will inevitably outcompete the other, while natural selection is invoked to explain how species adapt and persist despite such pressures.

To expose the inconsistency, the authors turn to the logistic equation, the workhorse of population dynamics. The logistic model describes the change in population size N over time t with two parameters: the intrinsic growth rate r and the carrying capacity K. In its usual form, the equation assumes that r and K are species‑specific constants that encapsulate all aspects of resource use, mortality, and reproduction. The authors point out that this formulation effectively treats each species as a “self‑replicating unit” with a fixed replication rate, analogous to a simple chemical reaction A → 2A. In this analogy, the flow of energy through an individual is the “reaction pathway,” and the pattern of that flow defines the species’ identity.

The central hypothesis of the paper is that biodiversity can be sustained only when different species share identical energy‑flow patterns. Under this “energy‑flow equivalence” assumption, two species are mathematically indistinguishable in the logistic framework; consequently, the competition term that would normally drive exclusion disappears, allowing stable coexistence. Conversely, if the energy‑flow patterns differ, the logistic model predicts that the species with the higher effective r (or the one that better matches K) will dominate, leading to the extinction of the competitor.

To test the hypothesis, the authors construct two sets of numerical simulations. In the first set, two hypothetical species are assigned identical r, K, and energy‑flow parameters. The simulations show that both populations converge to a stable equilibrium where each maintains a constant proportion of the total community, confirming the theoretical prediction of coexistence under energy‑flow equivalence. In the second set, the species share the same r and K but are given distinct energy‑flow patterns, which the authors model as a differential conversion efficiency in the logistic term. Here, the simulations produce the classic competitive exclusion outcome: one population rapidly declines to zero while the other approaches the carrying capacity.

The discussion emphasizes that the logistic equation, as traditionally employed, ignores the stochastic variation and selective pressures that are central to modern evolutionary theory. By treating r and K as fixed averages, the model effectively removes the very mechanism—mutation, differential fitness, and selection—that natural selection relies upon. The authors argue that to integrate natural selection with population dynamics, the model must be expanded to include explicit variance components, perhaps through stochastic differential equations or individual‑based models that track genotype frequencies.

Moreover, the authors contend that the “identical energy‑flow” condition is biologically implausible for most real‑world communities. Empirical studies show that even closely related species differ in metabolic pathways, foraging strategies, and microhabitat use, all of which translate into distinct energy‑flow signatures. Therefore, the paper concludes that the coexistence predicted by the logistic model under the authors’ assumptions is a highly constrained, perhaps theoretical, scenario rather than a general explanation for biodiversity.

In the final section, the authors call for a new theoretical framework that can simultaneously accommodate competition, resource limitation, and the evolutionary processes of mutation and selection. They suggest that future work should focus on quantifying species‑specific energy‑flow patterns, incorporating them as dynamic parameters in population models, and testing the resulting predictions against long‑term field data. By highlighting the incompatibility between a widely used kinetic model and the core tenets of natural selection, the paper aims to stimulate a re‑examination of the mathematical foundations of ecological and evolutionary theory.