Ecological theory of mutualism: Robust patterns of stability and thresholds in two-species population models

Mutualisms are ubiquitous in nature, provide important ecosystem services, and involve many species of interest for conservation. Theoretical progress on the population dynamics of mutualistic interactions, however, comparatively lagged behind that of trophic and competitive interactions, leading to the impression that ecologists still lack a generalized framework to investigate the population dynamics of mutualisms. Yet, over the last 90 years, abundant theoretical work has accumulated, ranging from abstract to detailed. Here, we review and synthesize historical models of two-species mutualisms. We find that population dynamics of mutualisms are qualitatively robust across derivations, including levels of detail, types of benefit, and inspiring systems. Specifically, mutualisms tend to exhibit stable coexistence at high density and destabilizing thresholds at low density. These dynamics emerge when benefits of mutualism saturate, whether due to intrinsic or extrinsic density-dependence in intraspecific processes, interspecific processes, or both. We distinguish between thresholds resulting from Allee effects, low partner density, and high partner density, and their mathematical and conceptual causes. Our synthesis suggests that there exists a robust population dynamic theory of mutualism that can make general predictions.

Gene drives and population persistence vs elimination: the impact of spatial structure and inbreeding at low density

Synthetic gene drive constructs are being developed to control disease vectors, invasive species, and other pest species. In a well-mixed random mating population a sufficiently strong gene drive is expected to eliminate a target population, but it is not clear whether the same is true when spatial processes play a role. In species with an appropriate biology it is possible that drive-induced reductions in density might lead to increased inbreeding, reducing the efficacy of drive, eventually leading to suppression rather than elimination, regardless of how strong the drive is. To investigate this question we analyse a series of explicitly solvable stochastic models considering a range of scenarios for the relative timing of mating, reproduction, and dispersal and analyse the impact of two different types of gene drive, a Driving Y chromosome and a homing construct targeting an essential gene. We find in all cases a sufficiently strong Driving Y will go to fixation and the population will be eliminated, except in the one life history scenario (reproduction and mating in patches followed by dispersal) where low density leads to increased inbreeding, in which case the population persists indefinitely, tending to either a stable equilibrium or a limit cycle. These dynamics arise because Driving Y males have reduced mating success, particularly at low densities, due to having fewer sisters to mate with. Increased inbreeding at low densities can also prevent a homing construct from eliminating a population. For both types of drive, if there is strong inbreeding depression, then the population cannot be rescued by inbreeding and it is eliminated. These results highlight the potentially critical role that low-density-induced inbreeding and inbreeding depression (and, by extension, other sources of Allee effects) can have on the eventual impact of a gene drive on a target population.

Periodic Solution of A Delayed Intraguild Predation Impulsive System with Strong Allee Effect

With periodic coefficients and strong Allee effects, we establish a delayed intraguild predation impulsive model. We obtain a set of sufficient conditions for the existence of positive periodic solution of the model using Mawhin’s continuation theorem and analysis techniques. Finally, we identify the effectiveness of the theoretical results through some numerical simulations.

Bifurcation analysis in a predator–prey model with strong Allee effect

In this paper, strong Allee effects on the bifurcation of the predator–prey model with ratio-dependent Holling type III response are considered, where the prey in the model is subject to a strong Allee effect. The existence and stability of equilibria and the detailed behavior of possible bifurcations are discussed. Specifically, the existence of saddle-node bifurcation is analyzed by using Sotomayor’s theorem, the direction of Hopf bifurcation is determined, with two bifurcation parameters, the occurrence of Bogdanov–Takens of codimension 2 is showed through calculation of the universal unfolding near the cusp. Comparing with the cases with a weak Allee effect and no Allee effect, the results show that the Allee effect plays a significant role in determining the stability and bifurcation phenomena of the model. It favors the coexistence of the predator and prey, can lead to more complex dynamical behaviors, not only the saddle-node bifurcation but also Bogdanov–Takens bifurcation. Numerical simulations and phase portraits are also given to verify our theoretical analysis.

Abundance And Active Patch Selection Modulate Reproductive Connectivity And Fitness of Pea Crabs Living On Sand Dollars

Connectivity is paramount for population stability, but the mechanisms underlying the distribution of populated patches and how they affect reproductive connectivity and individual fitness remain elusive. Here, we mapped the distribution of sand dollars – as habitat patches for obligate-commensal pea crabs – at several sites. At occupied patches, we assessed whole-crab population structure and the fitness of ovigerous females. While sand-dollar supply did not limit the size of crab populations, overall crab abundance limited reproductive connectivity and the potential for offspring production. However, except for sites of extremely low and high connectivity, crab aggregations at sand-dollar clusters countervailed the overall random distribution of sand-dollar populations, greatly enhancing the reproductive potential of whole-crab populations. Crab interactions, likely controlled by larger females, added to reproductive connectivity by increasing the frequency of mating pairs in hosts. Differently from the population-level case, effects of crab abundance on individual fitness were dual and only detectable when abundance was lowest (positive) or highest (negative), so that fitness remained high at intermediate crab abundance, decreasing when it became either too low (e.g. Allee effects) or too high (e.g. energetic costs of intraspecific competition). This study indicates that connectivity may affect different levels of biological organization in specific ways.

A mathematical modelling framework for the regulation of intra-cellular OCT4 in human pluripotent stem cells

Human pluripotent stem cells (hPSCs) have the potential to differentiate into all cell types, a property known as pluripotency. A deeper understanding of how pluripotency is regulated is required to assist in controlling pluripotency and differentiation trajectories experimentally. Mathematical modelling provides a non-invasive tool through which to explore, characterise and replicate the regulation of pluripotency and the consequences on cell fate. Here we use experimental data of the expression of the pluripotency transcription factor OCT4 in a growing hPSC colony to develop and evaluate mathematical models for temporal pluripotency regulation. We consider fractional Brownian motion and the stochastic logistic equation and explore the effects of both additive and multiplicative noise. We illustrate the use of time-dependent carrying capacities and the introduction of Allee effects to the stochastic logistic equation to describe cell differentiation. We conclude both methods adequately capture the decline in OCT4 upon differentiation, but the Allee effect model has the advantage of allowing differentiation to occur stochastically in a sub-set of cells. This mathematical framework for describing intra-cellular OCT4 regulation can be extended to other transcription factors and developed into predictive models.

Autocrine signaling explains the emergence of Allee effects in cancer cell populations

In many human cancers, the rate of cell growth depends crucially on the size of the tumor cell population. Low, zero, or negative growth at low population densities is known as the Allee effect; this effect has been studied extensively in ecology, but so far lacks a good explanation in the cancer setting. Here, we formulate and analyze an individual-based model of cancer, in which cell division rates are increased by the local concentration of an autocrine growth factor produced by the cancer cells themselves. We show, analytically and by simulation, that autocrine signaling suffices to cause both strong and weak Allee effects. Whether low cell densities lead to negative (strong effect) or reduced (weak effect) growth rate depends directly on the ratio of cell death to proliferation, and indirectly on cellular dispersal. Our model is consistent with experimental observations of brain tumor cells grown at different densities. We propose that further studying and quantifying population-wide feedback, impacting cell growth, will be central for advancing our understanding of cancer dynamics and treatment, potentially exploiting Allee effects for therapy.