The impact of predator dormancy on the population dynamics of phytoplankton-zooplankton in freshwater ecosystems is investigated using a simple model including dormancy, a strategy to avoid extinction. In addition to recently reported chaos-mediated mixed-mode oscillations, as the carrying capacity grows, we find surprisingly wide phases of nonchaos-mediated mixed-mode oscillations to be present well before the onset of chaos in the system. Nonchaos-mediated cascades display spike-adding sequences, while chaos-mediated cascades show spike-doubling. A host of braided periodic phases with exotic shapes is found embedded in a region of control parameters dominated by chaotic oscillations. We describe the organization of these complicated phases and show how they are interconnected and how their complexity unfolds as control parameters change. The novel nonchaos-mediated phases are found to be large and stable, even for low carrying capacity.

1.
M. L.
Rosenzweig
and
R. H.
MacArthur
, “
Graphical representation and stability conditions of predator-prey interactions
,”
Am. Nat.
97
,
209
223
(
1963
).
2.
M. L.
Rosenzweig
, “
Paradox of enrichment: Destabilization of exploitation ecosystems in ecological time
,”
Science
171
,
385
387
(
1971
).
3.
M.
Kuwamura
,
T.
Nakazawa
, and
T.
Ogawa
, “
A minimum model of prey-predator system with dormancy of predators and the paradox of enrichment
,”
J. Math. Biol.
58
,
459
479
(
2009
).
4.
C. X. J.
Jensen
and
L. R.
Ginzburg
, “
Paradox or theoretical failures? The jury is still out
,”
Ecol. Model.
188
,
3
14
(
2005
).
5.
M.
Kuwamura
and
H.
Chiba
, “
Mixed-mode oscillations and chaos in a prey-predator system with dormancy of predators
,”
Chaos
19
,
043121
(
2009
).
6.
V.
Alekseev
and
W.
Lampert
, “
Maternal control of resting-egg production in Dapnia
,”
Nature
414
,
899
901
(
2001
).
7.
M.
Gyllström
and
K.-A.
Hansson
, “
Dormancy in freshwater zooplankton: Induction, termination and the importance of benthic-pelagic coupling
,”
Aquat. Sci.
66
,
274
295
(
2004
).
8.
N. G.
Hairston
, Jr.
,
A. M.
Hansen
, and
W. R.
Schaffner
, “
The effect of diapause emergence on the seasonal dynamics of a zooplankton assemblage
,”
Freshwater Biol.
45
,
133
145
(
2000
).
9.
C.
Ricci
, “
Dormancy patterns in rorifers
,”
Hydrobiologia
446
,
1
11
(
2001
).
10.
E.
McCauley
,
R. M.
Nisbet
,
W. W.
Murdoch
,
A. M.
de Roos
, and
W. S. C.
Gurney
, “
Large-amplitude cycles of Daphnia and its algal prey in enriched environments
,”
Nature
402
,
653
656
(
1999
).
11.
M. J. B.
Hauser
and
J. A. C.
Gallas
, “
Nonchaos-mediated mixed-mode oscillations in an enzyme reaction system
,”
J. Phys. Chem. Lett.
5
,
4187
4193
(
2014
).
12.
M. R.
Gallas
and
J. A. C.
Gallas
, “
Nested arithmetic progressions of oscillatory phases in Olsen's enzyme reaction model
,”
Chaos
25
,
064603
(
2015
).
13.
J. G.
Freire
,
M. R.
Gallas
, and
J. A. C.
Gallas
, “
Chaos-free oscillations
,”
Europhys. Lett.
118
,
38003
(
2017
).
14.
J. G.
Freire
,
M. R.
Gallas
, and
J. A. C.
Gallas
, “
Stability mosaics in a forced Brusselator: Auto-organization of oscillations in control parameter space
,”
Eur. Phys. J. Special Topics
226
,
1987
1995
(
2017
).
15.
X.
Rao
,
Y.
Chu
,
Lu-Xu
,
Y.
Chang
, and
J.
Zhang
, “
Fractal structures in centrifugal flywheel governor system
,”
Commun. Nonlinear Sci. Numer. Simul.
50
,
330
339
(
2017
).
16.
J. G.
Freire
,
T.
Pöschel
, and
J. A. C.
Gallas
, “
Stern-Brocot trees in spiking and bursting of sigmoidal maps
,”
Europhys. Lett.
100
,
48002
(
2012
).
17.
J. G.
Freire
,
R. J.
Field
, and
J. A. C.
Gallas
, “
Relative abundance and structure of chaotic behavior: The nonpolynomial Belousov-Zhabotinsky reaction kinetics
,”
J. Chem. Phys.
131
,
044105
(
2009
).
18.
L.
Junges
and
J. A. C.
Gallas
, “
Intricate routes to chaos in the Mackey-Glass delayed feedback system
,”
Phys. Lett. A
376
,
2109
2116
(
2012
).
19.
J. A. C.
Gallas
, “
Spiking systematics in some CO2 laser models
,”
Adv. At., Mol., Opt. Phys.
65
,
127
191
(
2016
).
20.
M.
Kuwamura
, “
Turing instabilities in prey-predator systems with dormancy of predators
,”
J. Math. Biol.
71
,
125
149
(
2015
).
21.
R.
Vitolo
,
P.
Glendinning
, and
J. A. C.
Gallas
, “
Global structure of periodicity hubs in Lyapunov phase diagrams of dissipative flows
,”
Phys. Rev. E
84
,
016216
(
2011
).
22.
A. A.
Andronov
,
A. A.
Vitt
, and
S. E.
Khaikin
,
Theory of Oscillations
(
Dover
,
New York
,
1966
), translation of the 1937 Russian original.
23.
P.
Glendinning
, “
Bifurcations near homoclinic orbits with symmetry
,”
Phys. Lett. A
103
,
163
166
(
1984
).
24.
R.
Tchitnga
,
T.
Nguazon
,
P. H. L.
Fotso
, and
J. A. C.
Gallas
,
IEEE Trans. Circuit Syst. II
63
,
239
243
(
2016
).
25.
C.
Bonatto
and
J. A. C.
Gallas
, “
Periodicity hub and nested spirals in the phase diagram of a simple resistive circuit
,”
Phys. Rev. Lett.
101
,
054101
(
2008
).
26.
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: Auto-organization of oscillatory phases in parameter space
,”
Eur. Phys. J. Special Topics
223
,
2131
2144
(
2014
).
27.
E.
Benincà
,
J.
Huisman
,
R.
Heerkloss
,
K. D.
Jöhnk
,
P.
Branco
,
E. H.
Van Nes
,
M.
Scheffer
, and
S. P.
Ellner
, “
Chaos in a long-term experiment with a plankton community
,”
Nature
451
,
822
826
(
2008
).
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