Abstract
Traditional population models that include predator–prey interactions attribute demographic changes directly to predation-related effects. However, predator-induced fear in prey has increasingly been recognised as an important factor shaping population dynamics. In this study, we propose a cubic population model in which fear acts through two distinct functional channels for a single-species population exhibiting the Allee effect. In this model, fear reduces the intrinsic growth rate through a multiplicative suppression mechanism while also playing an integrated role in modulating the growth and interaction dynamics by rescaling the saturation structure of the Holling type III interaction term. The stochastic extension of the model is described by a Langevin formalism containing correlated additive and multiplicative Gaussian noise, and the steady state probability distribution (SSPD) is analytically obtained using the corresponding Fokker–Planck equation. The analytical solution is validated by numerical simulations. The SSPD reveals both noise-induced transitions and fear-controlled regime changes between low- and high-density states, with the dual-channel structure producing structural competition and non-monotonic changes in the distribution. These are analysed through phenomenological bifurcation (P-bifurcation) diagrams and three-dimensional distribution surfaces. Additionally, statistical properties, parameter sensitivity, and escape dynamics are investigated through normalised moments, Fisher information, and mean first-passage time (MFPT) calculations. Notably, our model treats fear as an independent control parameter and provides a theoretical mechanism capable of generating qualitatively contrasting population responses across parameter regimes, while also offering a framework for examining critical transition dynamics of potential relevance to conservation and management.
Funding
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
Appendix A: Numerical parameter settings
See Table 3.
Appendix B: Explicit form of the SSPD
The closed-form expression below is written for (a_2>0), which corresponds to (f>0) in the principal model. For (a_2=0), the stationary density is obtained directly from the general representation in Eq. (23).
The explicit steady state probability distribution used in Eq. (26) is
with the normalisation condition
The exponent function is given by
The auxiliary quantities are
and
Appendix C: Numerical validation of the Kramers approximation
The accuracy of the Kramers approximation was assessed by comparing Eq. (40) with the integral representation in Eq. (38), with the exponential factor obtained by direct numerical quadrature of (A(x)/B(x)). Over the parameter ranges of Fig. 12, the relative difference remains within (4%) in all panels with negative noise correlation and reaches at most about (11%) in the positive-correlation panels at the largest (beta) shown. The least favourable case occurs for (gamma =0.6), (a_1=0.1), (f=a_2=1.4), (alpha =0.02), and (beta =0.01). At this point, a direct Milstein simulation using 3000 trajectories initiated at (x_{s_2}) and absorbed at (x_{textrm{LD}}=x_{s_1}/2), with (Delta t=5times 10^{-4}), gives (T=325pm 6) (mean ± standard error), compared with (324) from Eq. (38) and (288) from Eq. (40). The dependence on the target location is weak: as (x_{textrm{LD}}) is varied between (0.3x_{s_1}) and (0.7x_{s_1}), (ln T) changes by less than (0.02) in the negative-correlation panels and by less than (0.2) in the positive-correlation panels, which is small on the logarithmic scale of Fig. 12.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Reprints and permissions
About this article
Cite this article
Gültekin, Ö., Muradli, M. Dual fear mechanisms shaping stochastic population dynamics under the Allee effect.
Sci Rep (2026). https://doi.org/10.1038/s41598-026-70008-4
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s41598-026-70008-4
Keywords
- Fear effect
- Allee effect
- Stochastic population dynamics
- Fokker–Planck equation
- Steady state probability distribution
- Noise-induced transitions
- Mean first-passage time
Source: Ecology - nature.com
