Vol.:(0123456789) 1 3 Modeling Earth Systems and Environment https://doi.org/10.1007/s40808-020-01051-x ORIGINAL ARTICLE Noise‑induced control of environmental fluctuations in a three‑species predator–prey model Kalyan Das 1  · M. N. Srinivash 2  · M. Humayun Kabir 3  · M. Osman Gani 3 Received: 16 September 2020 / Accepted: 6 November 2020 © Springer Nature Switzerland AG 2020 Abstract A three-species ecological model is proposed that comprises one prey and two predators (low predator and top predator) to understand the persistence and/or extinction of prey and predators under a suitable environment. The realistic ecological system is typically inhomogeneous in terms of spatiotemporal variations of population density. With this motivation, white noise is introduced to the system to conceive the impact of environmental heterogeneity on the interaction of prey and preda- tors. The boundedness of solutions of the system is presented. The condition for the local stability of the deterministic model is established. The stochastic analysis of the system suggests the impression of the environmental noise on the dynamical behaviour of solutions. Numerical results reveal that the proposed system exhibits an oscillatory pattern of solutions. It is shown that the white noise in the low predator has a dominating role in comparison with other noises to keep the oscilla- tion in order. Moreover, to investigate the spatial dependency of the population density, diffusion term is introduced to the system that also displays oscillatory behaviour of the solutions. The results also confirm that the population density of prey, low predator, and top predator have a biological control in the oscillatory behaviour. In the presence of diffusion and white noise, numerical simulation illustrates the sensitivity of the diffusion coefficients and the amplitude of noises. Numerical results lead to a reasonable agreement with the analytical findings in both non-spatial and spatial systems. Keywords Prey–predator · Food chain · Environmental noise · Oscillations · Diffusion · Numerical simulation Mathematics Subject Classification 60H40 · 34M10 · 60H10 · 65L07 · 65M22 · 92D25 Introduction In ecological systems, prey–predator plays a vital role as they are participating at the base level of the food chain. In particular, prey–predator can carry out feeding, which forms the base of the food chains in any ecological com- partments that may lead to failure and even cause an imbal- ance in environments (Pielou et al. 1969; Pielou 1974). But the process by which the imbalance occurs is not yet clearly understand in reality. Hence, an understanding of the dynamic mechanism of changes for any prey–predator popu- lation with the impact of environmental noise and diffusion becomes much more significant to the researchers (Manna et al. 2020). Over the last four decades, experimental as well as mathematical ecologists have paid increasing attention to the phenomena of prey–predator interaction in any ecologi- cal compartments (Heck and Thoman 1981; Kawasaki and Teramoto 1979). And researchers are attempting to explain the dynamic mechanism of the prey–predator model by * M. Humayun Kabir hkabirju@gmail.com Kalyan Das daskalyan27@gmail.com M. N. Srinivash mnsrinivaselr@gmail.com M. Osman Gani osmangani2@juniv.edu 1 Department of Basic and Applied Sciences, National Institute of Food Technology Entrepreneurship and Management, Kundli, Haryana 131 028, India 2 Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore 632 014, India 3 Department of Mathematics, Jahangirnagar University, Dhaka 1342, Bangladesh