A Nonlinear Dynamical Model of the Interaction between Soil Degradation, Agricultural Productivity and Food Security: Well-posedness, Threshold Dynamics, Hopf Bifurcation and Management Scenarios

Publication Details

Publication Date

10/11/2026

Volume / Issue

Vol 1, Issue 6 (2026)

Article No.

035

Journal

British Journal of Contemporary Research

DOI

https://doi.org/10.67693/BJCR-MCLR636Q

Received

15 Oct 2026

Views

19

Downloads

11

Abstract

We develop and analyse a four-dimensional nonlinear ordinary differential equation model for normalised soil quality S, agricultural productivity A, food security F and soil-degradation pressure D, closed by the hypothesis that low food security raises production pressure and hence degradation. We prove positivity, boundedness and global existence on an explicit invariant region. The equilibrium problem is reduced to one strictly monotone scalar equation, which gives a sharp threshold: a unique positive equilibrium E* exists if and only if the productivity number RA=ra++/ exceeds one, and when RA1 the productivity-free equilibrium is globally asymptotically stable. At E* we derive the Jacobian and characteristic polynomial, prove that all four coefficients are positive for every admissible parameter set, and conclude that E* can lose stability only through a Hopf bifurcation, with an explicit Routh–Hurwitz criterion. A centre-manifold calculation shows that the transcritical bifurcation at RA=1 is forward. Numerical work, with hypothetical parameter values, gives eigenvalues and eigenvectors for four management scenarios, local and global sensitivity rankings, and a verified supercritical Hopf bifurcation in a stress-test parameter set. In the illustrative baseline, combining soil restoration with reduced degradation intensity raises equilibrium soil quality by about 110%, productivity by about 60% and food security by about 27%, with sub-additive gains. Global stability of E* is not claimed, and the results are not calibrated to data.

License

This article is published under the Creative Commons Attribution 4.0 International License . Free to read, share, and adapt with attribution.