In this work, we investigate the dynamics of a discrete-time prey–predator model considering a prey reproductive response as a function of the predation risk, with the prey population growth factor governed by two parameters. The system can evolve toward scenarios of mutual or only of predators extinction, or species coexistence. We analytically show all different types of equilibrium points depending on the ranges of growth parameters. By numerical study, we find the occurrence of quasiperiodic, chaotic, and hyperchaotic behaviors. Our analytical results are corroborated by the numerical ones. We highlight Arnold tongue-like periodic structures organized according to the Farey sequence, as well as pairs of twin shrimps connected by two links. The mathematical model captures two possible prey responsive strategies, decreasing or increasing the reproduction rate under predatory threat. Our results support that both strategies are compatible with the populations coexistence and present rich dynamics.
REFERENCES
1.
T. R.
Malthus
, An Essay on the Principle of Population
(Reeves and Turner
, London
, 1878
).2.
A. J.
Lotka
, Analytical Theory of Biological Populations
(Springer Science & Business Media
, 1998
).3.
N.
Bacaër
, A Short History of Mathematical Population Dynamics
(Springer
, 2011
), Vol. 618.4.
5.
P.-F.
Verhulst
, “Recherches mathématiques sur la loi d’accroissement de la population
,” Mém. l’Acad. r. Belg.
18
, 1
–40
(1845
).6.
P.-F.
Verhulst
, Deuxième Mémoire sur la Loi d’accroissement de la Population
(Hayez
, 1847
), Vol. 269.7.
D.
Tilman
, “The importance of the mechanisms of interspecific competition
,” Am. Nat.
129
, 769
–774
(1987
). 8.
P.
Chesson
and J. J.
Kuang
, “The interaction between predation and competition
,” Nature
456
, 235
–238
(2008
). 9.
K.
Burns
and P.
Lester
, “Competition and coexistence in model populations,” in Encyclopedia of Ecology (Academic Press, Oxford, 2008), pp. 701–707.10.
W. J.
Ripple
and R. L.
Beschta
, “Trophic cascades in Yellowstone: The first 15 years after wolf reintroduction
,” Biol. Conserv.
145
, 205
–213
(2012
). 11.
A. J.
Lotka
, “Analytical note on certain rhythmic relations in organic systems
,” Proc. Natl. Acad. Sci. U.S.A.
6
, 410
–415
(1920
). 12.
13.
V.
Volterra
, “Variazioni e fluttuazioni del numero d’individui in specie animali conviventi
,” Memor. Accad. Lincei, Ser.
6
, 31
–113
(1926
).14.
V.
Volterra
, “Fluctuations in the abundance of a species considered mathematically
,” Nature
118
, 558
–560
(1926
). 15.
J.
Hofbauer
and K.
Sigmund
, “Lotka–Volterra equations for predator–prey systems,” in Evolutionary Games and Population Dynamics (Cambridge University Press, 1998), pp. 11–21.16.
Y.
Takeuchi
, Global Dynamical Properties of Lotka-Volterra Systems
(World Scientific
, 1996
).17.
J. G.
Freire
, M. R.
Gallas
, and J. A. C.
Gallas
, “Impact of predator dormancy on prey-predator dynamics
,” Chaos
28
, 053118
(2018
). 18.
V. A.
Kuznetsov
, I. A.
Makalkin
, M. A.
Taylor
, and A. S.
Perelson
, “Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis
,” Bull. Math. Biol.
56
, 295
–321
(1994
). 19.
L. G.
De Pillis
and A.
Radunskaya
, “The dynamics of an optimally controlled tumor model: A case study
,” Math. Comput. Model.
37
, 1221
–1244
(2003
). 20.
M. R.
Gallas
, M. R.
Gallas
, and J. A. C.
Gallas
, “Distribution of chaos and periodic spikes in a three-cell population model of cancer
,” Eur. Phys. J. Spec. Top.
223
, 2131
–2144
(2014
).21.
R. M.
May
, “Simple mathematical models with very complicated dynamics
,” Nature
261
, 459
–467
(1976
). 22.
M. J.
Feigenbaum
, “Universal behavior in nonlinear systems
,” Physica D
7
, 16
–39
(1983
). 23.
J. A. C.
Gallas
, “Nonlinear dependencies between sets of periodic orbits
,” Europhys. Lett.
47
, 649
(1999
). 24.
J. E. S.
Socolar
, “Chaos,” in Encyclopedia of Physical Science and Technology, 3rd ed., edited by R. A. Meyers (Academic Press, New York, 2003), pp. 637–665.25.
J. A. C.
Gallas
, “Preperiodicity and systematic extraction of periodic orbits of the quadratic map
,” Int. J. Mod. Phys. C
31
, 2050174
(2020
). 26.
E. N.
Lorenz
, “The problem of deducing the climate from the governing equations
,” Tellus
16
, 1
–11
(1964
). 27.
S. M.
Ulam
and J.
Von Neumann
, “On combination of stocastic deterministic processes
,” Bull. AMS
53
, 1120
(1947
).28.
M.
Gyllenberg
, G.
Söderbacka
, and S.
Ericsson
, “Does migration stabilize local population dynamics? Analysis of a discrete metapopulation model
,” Math. Biosci.
118
, 25
–49
(1993
). 29.
A.
Hastings
, “Complex interactions between dispersal and dynamics: Lessons from coupled logistic equations
,” Ecology
74
, 1362
–1372
(1993
). 30.
A. L.
Lloyd
, “The coupled logistic map: A simple model for the effects of spatial heterogeneity on population dynamics
,” J. Theor. Biol.
173
, 217
–230
(1995
). 31.
Y. A.
Kuznetsov
, Elements of Applied Bifurcation Theory
, 4th ed. (Springer
, 2023
), Vol. 112.32.
P. C.
Rech
, M. W.
Beims
, and J. A. C.
Gallas
, “Generation of quasiperiodic oscillations in pairs of coupled maps
,” Chaos, Solitons Fractals
33
, 1394
–1410
(2007
). 33.
G. C.
Layek
and N. C.
Pati
, “Organized structures of two bidirectionally coupled logistic maps
,” Chaos
29
, 093104
(2019
). 34.
35.
G. I.
Bischi
and F.
Tramontana
, “Three-dimensional discrete-time Lotka–Volterra models with an application to industrial clusters
,” Commun. Nonlinear Sci. Numer. Simul.
15
, 3000
–3014
(2010
). 36.
Q.
Din
, “Dynamics of a discrete Lotka-Volterra model
,” Adv. Differ. Equ.
2013
, 1
–13
.37.
A. Q.
Khan
and M. N.
Qureshi
, “Global dynamics and bifurcations analysis of a two-dimensional discrete-time Lotka-Volterra model
,” Complexity
2018
, 7101505
. 38.
R.
Shine
and T.
Madsen
, “Prey abundance and predator reproduction: Rats and pythons on a tropical Australian floodplain
,” Ecology
78
, 1078
–1086
(1997
).39.
N. P.
Anders
, “Predator behaviour and prey density: Evaluating density-dependent intraspecific interactions on predator functional responses
,” J. Anim. Ecol.
70
, 14
–19
(2001
).40.
M. E.
Solomon
, “The natural control of animal populations
,” J. Anim. Ecol.
18
, 1
–35
(1949
). 41.
V.
Křivan
, “Prey–predator models,” in Encyclopedia of Ecology, edited by S. E. Jørgensen and B. D. Fath (Academic Press, Oxford, 2008), pp. 2929–2940.42.
M. J.
Cherry
, K. E.
Morgan
, B. T.
Rutledge
, L. M.
Conner
, and R. J.
Warren
, “Can coyote predation risk induce reproduction suppression in white-tailed deer?
” Ecosphere
7
, e01481
(2016
). 43.
S.
Creel
and D.
Christianson
, “Relationships between direct predation and risk effects
,” Trends Ecol. Evol.
23
, 194
–201
(2008
).44.
K.
Norrdahl
and E.
KorpimÄki
, “The impact of predation risk from small mustelids on prey populations
,” Mammal Rev.
30
, 147
–156
(2000
). 45.
G. D.
Ruxton
and S. L.
Lima
, “Predator–induced breeding suppression and its consequences for predator–prey population dynamics
,” Proc. R. Soc. Lond., Ser. B
264
, 409
–415
(1997
). 46.
M.
Danca
, S.
Codreanu
, and B.
Bako
, “Detailed analysis of a nonlinear prey-predator model
,” J. Biol. Phys.
23
, 11
(1997
). 47.
C. S.
Holling
, “The components of predation as revealed by a study of small-mammal predation of the European pine sawfly
,” Can. Entomol.
91
, 293
–320
(1959
). 48.
J.
Dawes
and M.
Souza
, “A derivation of Holling’s type I, II and III functional responses in predator–prey systems
,” J. Theor. Biol.
327
, 11
–22
(2013
). 49.
50.
K.
Alligood
, T.
Sauer
, and J.
Yorke
, Chaos: An Introduction to Dynamical Systems, Textbooks in Mathematical Sciences (Springer, New York, 1996).51.
J.-P.
Eckmann
and D.
Ruelle
, “Ergodic theory of chaos and strange attractors
,” Rev. Mod. Phys.
57
, 617
(1985
). 52.
G.
Benettin
, L.
Galgani
, A.
Giorgilli
, and J.-M.
Strelcyn
, “Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them. Part 1: Theory
,” Meccanica
15
, 9
–20
(1980
). 53.
G.
Benettin
, L.
Galgani
, A.
Giorgilli
, and J.-M.
Strelcyn
, “Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them. Part 2: Numerical application
,” Meccanica
15
, 21
–30
(1980
). 54.
T.
Kapitaniak
, Y.
Maistrenko
, and S.
Popovych
, “Chaos-hyperchaos transition
,” Phys. Rev. E
62
, 1972
(2000
). 55.
G. H.
Hardy
and E. M.
Wright
, An Introduction to the Theory of Numbers
(Oxford University Press
, 1979
).56.
“Small denominators. I. Mapping of the circumference onto itself,” in Collected Works: Representations of Functions, Celestial Mechanics and KAM Theory, 1957–1965, edited by A. B. Givental, B. A. Khesin, J. E. Marsden, A. N. Varchenko, V. A. Vassiliev, O. Y. Viro, and V. M. Zakalyukin (Springer, Berlin, 2009), pp. 152–223.
57.
P. C.
Rech
, “Organization of the periodicity in the parameter-space of a glycolysis discrete-time mathematical model
,” J. Math. Chem.
57
, 632
–637
(2019
). 58.
J. A.
Gallas
, “Structure of the parameter space of the Hénon map
,” Phys. Rev. Lett.
70
, 2714
(1993
). 59.
R.
Vitolo
, P.
Glendinning
, and J. A.
Gallas
, “Global structure of periodicity hubs in Lyapunov phase diagrams of dissipative flows
,” Phys. Rev. E
84
, 016216
(2011
). 60.
G. M.
Ramírez-Ávila
and J. A. C.
Gallas
, “How similar is the performance of the cubic and the piecewise-linear circuits of Chua?
” Phys. Lett. A
375
, 143
–148
(2010
).61.
S. L.
de Souza
, I. L.
Caldas
, R. L.
Viana
et al., “Multistability and self-similarity in the parameter-space of a vibro-impact system
,” Math. Probl. Eng.
2009
, 290356
(2009
).62.
S. L.
de Souza
, A. A.
Lima
, I. L.
Caldas
, R. O.
Medrano-T
, and Z. D. O.
Guimarães-Filho
, “Self-similarities of periodic structures for a discrete model of a two-gene system
,” Phys. Lett. A
376
, 1290
–1294
(2012
). 63.
C.
Bonatto
, J. C.
Garreau
, and J. A.
Gallas
, “Self-similarities in the frequency-amplitude space of a loss-modulated CO
laser
,” Phys. Rev. Lett.
95
, 143905
(2005
). © 2025 Author(s). Published under an exclusive license by AIP Publishing.
2025
Author(s)
You do not currently have access to this content.
