Using neural ordinary differential equations to predict complex ecological dynamics from population density data

Simple models have been used to describe ecological processes for over a century. However, the complexity of ecological systems makes simple models subject to modelling bias due to simplifying assumptions or unaccounted factors, limiting their predictive power. Neural ordinary differential equations (NODEs) have surged as a machine-learning algorithm that preserves the dynamic nature of the data (Chen et al. 2018 Adv. Neural Inf. Process. Syst.). Although preserving the dynamics in the data is an advantage, the question of how NODEs perform as a forecasting tool of ecological communities is unanswered. Here, we explore this question using simulated time series of competing species in a time-varying environment. We find that NODEs provide more precise forecasts than autoregressive integrated moving average (ARIMA) models. We also find that untuned NODEs have a similar forecasting accuracy to untuned long-short term memory neural networks and both are outperformed in accuracy and precision by empirical dynamical modelling . However, we also find NODEs generally outperform all other methods when evaluating with the interval score, which evaluates precision and accuracy in terms of prediction intervals rather than pointwise accuracy. We also discuss ways to improve the forecasting performance of NODEs. The power of a forecasting tool such as NODEs is that it can provide insights into population dynamics and should thus broaden the approaches to studying time series of ecological communities.

A mathematical model with nonlinear relapse: conditions for a forward-backward bifurcation

We constructed a Susceptible-Addicted-Reformed model and explored the dynamics of nonlinear relapse in the Reformed population. The transition from susceptible considered at-risk is modeled using a strictly decreasing general function, mimicking an influential factor that reduces the flow into the addicted class. The basic reproductive number is computed, which determines the local asymptotically stability of the addicted-free equilibrium. Conditions for a forward-backward bifurcation were established using the basic reproductive number and other threshold quantities. A stochastic version of the model is presented, and some numerical examples are shown. Results showed that the influence of the temporarily reformed individuals is highly sensitive to the initial addicted population.

How far to build it before they come? Analyzing the use of the Field of Dreams hypothesis in bull kelp restoration

In restoration ecology, the Field of Dreams hypothesis posits that restoration efforts that create a suitable environment could lead to the eventual recovery of the remaining aspects of the ecosystem through natural processes. Natural processes following partial restoration has led to ecosystem recovery in both terrestrial and aquatic systems. However, understanding the efficacy of a "Field of Dreams" approach requires a comparison of different approaches to partial restoration in terms of spatial, temporal, and ecological scale with what would happen given more comprehensive restoration efforts. We explore the relative effect of partial restoration and ongoing recovery on restoration efficacy with a dynamical model based on temperate rocky reefs in Northern California. We analyze our model for both the ability and rate of bull kelp forest recovery under different restoration strategies. We compare the efficacy of a partial restoration approach with a more comprehensive restoration effort by exploring how kelp recovery likelihood and rate change with varying intensities of urchin removal and kelp outplanting over different time periods and spatial scales. We find that, in the case of bull kelp forests, setting more favorable initial conditions for kelp recovery by implementing both urchin harvesting and kelp outplanting at the start of the restoration project has a bigger impact on the kelp recovery rate than applying restoration efforts through a longer period of time. Therefore, partial restoration efforts, in terms of spatial and temporal scale, can be significantly more effective when applied across multiple ecological scales in terms of both the capacity and rate for achieving the target outcomes.

Characterizing Long Transients in Consumer–Resource Systems With Group Defense and Discrete Reproductive Pulses

During recent years, the study of long transients has been expanded in ecological theory to account for shifts in long-term behavior of ecological systems. These long transients may lead to regime shifts between alternative states that resemble the dynamics of alternative stable states for a prolonged period of time. One dynamic that potentially leads to long transients is the group defense of a resource in a consumer–resource interaction. Furthermore, time lags in the population caused by discrete reproductive pulses have the potential to produce long transients, either independently or in conjunction to the transients caused by the group defense. In this work, we analyze the potential for long transients in a model for a consumer–resource system in which the resource exhibits group defense and reproduces in discrete reproductive pulses. This system exhibits crawl-by transients near the extinction and carrying capacity states of resource, and a transcritical bifurcation, under which a ghost limit cycle appears. We estimate the transient time of our system from these transients using perturbation theory. This work advances an understanding of how systems shift between alternate states and their duration of staying in a given regime and what ecological dynamics may lead to long transients.

The effect of colonization dynamics in competition for space in metacommunities

One of the main factors that determines habitat suitability for sessile and territorial organisms is the presence or absence of another competing individual in that habitat. This type of competition arises in populations occupying patches in a metacommunity. Previous studies have looked at this process using a continuous-time modeling framework, where colonizations and extinctions occur simultaneously. However, different colonization processes may be performed by different species, which may affect the metacommunity dynamics. We address this issue by developing a discrete-time framework that describes these kinds of metacommunity interactions, and we consider different colonization dynamics. To understand potential dynamics, we consider specific functional forms that characterize the colonization and extinction processes of metapopulations competing for space as their limiting factor. We then provide a mathematical analysis of the models generated by this framework, and we compare these results to what is seen in nature and in previous models.