One of the main challenges in ecology is determining the cause of population fluctuations. Both theoretical and empirical studies suggest that delayed density dependence instigates cyclic behavior in many populations, however underlying mechanisms through which this occurs are often difficult to determine and may vary within species. In this paper we consider single species population dynamics affected by the Allee effect coupled with discrete time delay. We use two different mathematical formulations of the Allee effect and analyze (both analytically and numerically) the role of time delay in different feedback mechanisms such as competition and cooperation. The bifurcation value of the delay (that results in the Hopf bifurcation) as a function of the strength of the Allee effect is obtained analytically. Interestingly, depending on the chosen delayed mechanism, even a large time delay may not necessarily lead to instability. We also show that, in case the time delay affects positive feedback (such as cooperation), the population dynamics can lead to self-organized formation of intermediate quasi-stationary states. Finally, we discuss ecological implications of our findings.

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Questions surrounding the prevalence of complex population dynamics form one of the central themes in ecology. Limit cycles and spatiotemporal chaos are examples that have been widely recognised theoretically, although their importance and applicability to natural populations remains debatable. The ecological processes underlying such dynamics are thought to be numerous, though there seems to be consent as to delayed density dependence being one of the main driving forces. Indeed, time delay is a common feature of many ecological systems and can signi cantly in uence population dynamics. In general, time delays may arise from interand intra-speci c trophic interactions or population structure, however in the context of single species populations they are linked to more intrinstic biological phenomena such as gestation or resource regeneration. In this paper, we consider theoretically the spatiotemporal dynamics of a single species population using two di erent mathematical formulations. Firstly, we revisit the di usive logistic equation in which the per capita growth is a function of some speci ed delayed argument. We then modify the model by incorporating a spatial convolution which results in a biologically more viable integro-di erential model. Using the combination of analytical and numerical techniques, we investigate the e ect of time delay on pattern formation. In particular, we show that for su ciently large values of time delay the system's dynamics are indicative to spatiotemporal chaos. The chaotic dynamics arising in the wake of a travelling population front can be preceded by either a plateau corresponding to dynamical stabilisation of the unstable equilibrium or by periodic oscillations.