PERIOD-DOUBLING AND NEIMARK–SACKER BIFURCATIONS IN A DISCRETE PREDATOR–PREY MAP WITH ALLEE EFFECT: CENTER MANIFOLD ANALYSIS, CHAOS AND HYBRID CONTROL
Abstract
Populations that reproduce in discrete generations and suffer a strong Allee effect can lose
stability in ways that continuous-time models miss. This paper studies the two-dimensional
map
xn+1 = xn exp[︁r(1 −xn)(xn −m)−ayn
]︁,
yn+1 = yn exp[︁ −d+bxn
]︁,
which couples a Ricker-type prey with a strong Allee threshold m ∈ (0,1) to a predator
with a linear response. We find all fixed points, prove that the interior fixed point exists
exactly when m < d/b < 1, and use the Jury conditions to obtain necessary and sufficient
conditions for its asymptotic stability. The interior fixed point loses stability through a flip
bifurcation as the prey growth rate r increases and through a Neimark–Sacker bifurcation
as the predator conversion rate b increases, and we give both critical values in closed form.
Center manifold reduction and normal-form coefficients determine the direction of each
bifurcation. The flip bifurcation is supercritical for some parameter values, where it starts a
period-doubling cascade to chaos with a positive largest Lyapunov exponent, and subcritical
for others, where the loss of stability leads directly to extinction of both species. In the
supercritical Neimark–Sacker case an attracting invariant closed curve appears, and the
largest Lyapunov exponent stays at zero. Bifurcation diagrams and Lyapunov exponents
agree with the analytical critical values. Finally, we apply a hybrid control that replaces the
mapF byαF+(1−α)idwithα ∈ (0,1). Itkeepsthefixedpoints, delays both bifurcations,
and in the computed examples restores the interior fixed point in the chaotic regime and
prevents extinction.
References
Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory, 3rd ed., Applied Mathematical
Sciences, vol. 112, Springer, New York, 2004.
J. Carr, Applications of Centre Manifold Theory, Applied Mathematical Sciences, vol. 35,
Springer, New York, 1981.
S. N. Elaydi, Discrete Chaos: With Applications in Science and Engineering, 2nd ed.,
Chapman & Hall/CRC, Boca Raton, 2008.
E. I. Jury and J. Blanchard, A stability test for linear discrete systems in table form,
Proceedings of the IRE, 49(12) (1961), 1947–1948.
R. J. Sacker, A new approach to the perturbation theory of invariant surfaces, Communica
tions on Pure and Applied Mathematics, 18(4) (1965), 717–732
A. Wolf, J. B. Swift, H. L. Swinney and J. A. Vastano, Determining Lyapunov exponents
from a time series, Physica D, 16(3) (1985), 285–317.
E. Ott, C. Grebogi and J. A. Yorke, Controlling chaos, Physical Review Letters, 64(11)
(1990), 1196–1199.
X. S. Luo, G. R. Chen, B. H. Wang, J. Q. Fang, Y. L. Zou and H. J. Quan, Hybrid control
of period-doubling bifurcation and chaos in discrete nonlinear dynamical systems, Chaos,
Solitons & Fractals, 18(4) (2003), 775–783.
F. Courchamp, T. Clutton-Brock and B. Grenfell, Inverse density dependence and the Allee
effect, Trends in Ecology & Evolution, 14(10) (1999), 405–410.
Z. Jing and J. Yang, Bifurcation and chaos in discrete-time predator–prey system, Chaos,
Solitons & Fractals, 27(1) (2006), 259–277
