Nullcline Geometry Constrains the Location of Oscillatory Instabilities in Planar Predator–Prey Systems

• E. Chan-López ^[1] ; A. Martín-Ruiz ^[2] ; V. Castellanos ^[1]
  1. [1] Universidad Juárez Autónoma de Tabasco

     Universidad Juárez Autónoma de Tabasco

     México

     
  2. [2] Universidad Nacional Autónoma de México

     Universidad Nacional Autónoma de México

     México

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 25, Nº 5, 2026
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • We prove that the critical structure of the prey nullcline constrains where oscillatory instabilities can emerge in planar predator–prey systems of Gause type. Under a predator self–damping hypothesis, the prey coordinate of every Hopf bifurcation point is confined to an ascending branch of the prey nullcline, between consecutive critical points. The mechanism is purely geometric: along the nullcline the diagonal Jacobian entry satisfies J11 = p(x) g (x), independently of the bifurcation parameter, while J22 < 0 at any coexistence equilibrium; at a critical point (g = 0) the trace is therefore strictly negative, so the critical points act as spectral barriers that preclude oscillatory onset. We illustrate the principle in three canonical families—Bazykin’s model (quadratic nullcline), a Leslie–Holling type IV system with harvesting (cubic nullcline), and the Crowley–Martin model with predator interference (rational nullcline)—for which we obtain closed–form Hopf loci. The mechanism persists in discrete time: for the forward Euler map with step size τ, the exact identity det(JG) − 1 = τ [tr(J ) + τ det(J )] shows that the Neimark–Sacker locus is strictly disjoint from the continuous Hopf locus, differs from it by the O(τ ) term τ det(J ), and crosses the spectral barrier only in the coarse–step regime. Predator saturation and interference thus not only bound the efficiency of interactions but also imprint a geometric architecture on the bifurcation landscape.

• Referencias bibliográficas
  • 1. Andrews, J.F.: A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates. Biotechnol. Bioeng. 10,...
  • 2. Bazykin, A.D.: Nonlinear Dynamics of Interacting Populations. World Scientific, Singapore (1998)
  • 3. Cheng, L., Zhang, L.: Bogdanov-Takens bifurcation of a Holling type IV predator-prey model with constant-effort harvesting. J. Inequal....
  • 4. Crowley, P.H., Martin, E.K.: Functional responses and interference within and between year classes of a dragonfly population. J. North...
  • 5. Dai, Y., Zhao, Y.: Hopf cyclicity and global dynamics for a predator-prey system of Leslie type with simplified Holling type IV functional...
  • 6. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York (1983)
  • 7. Hammoum, A., Sari, T., Yadi, K.: The Rosenzweig-MacArthur graphical criterion for a predator-prey model with variable mortality rate. Qual....
  • 8. Huang, J., Xia, X., Zhang, X., Ruan, S.: Bifurcation of codimension 3 in a predator-prey system of Leslie type with simplified Holling...
  • 9. Kot, M.: Elements of Mathematical Ecology. Cambridge University Press, Cambridge (2001)
  • 10. Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory, 3rd edn. Springer, New York (2004)
  • 11. Kuznetsov, Y.A., Meijer, H.G.E.: Numerical Bifurcation Analysis ofMaps: From Theory to Software. Cambridge University Press, Cambridge...
  • 12. Li, Y., Xiao, D.: Bifurcations of a predator-prey system of Holling and Leslie types. Chaos, Solitons Fractals 34, 606–620 (2007)
  • 13. Lu, M., Huang, J.: Global analysis in Bazykin’s model with Holling type II functional response and
  • predator competition. J. Differ. Equ. 280, 99–138 (2021) 14. Rosenzweig,M.L., MacArthur,R.H.: Graphical representation and stability conditions...
  • 15. Rosenzweig, M.L.: Paradox of enrichment: destabilization of exploitation ecosystems in ecological time. Science 171, 385–387 (1971)
  • 16. Shang, Z., Qiao, Y.: Multiple bifurcations in a predator-prey system of modified Holling and Leslie type with double Allee effect and...
  • 17. Zhang, M., Li, Z., Chen, F., et al.: Bifurcation analysis of a Leslie-Gower predator-prey model with Allee effect on predator and simplified...