The transient dynamics capture the time history in the behavior of a system before reaching an attractor. This paper deals with the statistics of transient dynamics in a classic tri-trophic food chain with bistability. The species of the food chain model either coexist or undergo a partial extinction with predator death after a transient time depending upon the initial population density. The distribution of transient time to predator extinction shows interesting patterns of inhomogeneity and anisotropy in the basin of the predator-free state. More precisely, the distribution shows a multimodal character when the initial points are located near a basin boundary and a unimodal character when chosen from a location far away from the boundary. The distribution is also anisotropic because the number of modes depends on the direction of the local of initial points. We define two new metrics, viz., homogeneity index and local isotropic index, to characterize the distinctive features of the distribution. We explain the origin of such multimodal distributions and try to present their ecological implications.
REFERENCES
1
A.
Hastings
, K. C.
Abbott
, K.
Cuddington
, T.
Francis
, G.
Gellner
, Y.-C.
Lai
, A.
Morozov
, S.
Petrovskii
, K.
Scranton
, and M.
Lou Zeeman
, “Transient phenomena in ecology
,” Science
361
(6406
), eaat6412
(2018
). 2
Y.-C.
Lai
and T.
Tél
, Transient Chaos: Complex Dynamics on Finite Time Scales
(Springer Science & Business Media
, 2011
), Vol. 173.3
M.
Dhamala
and Y.-C.
Lai
, “Controlling transient chaos in deterministic flows with applications to electrical power systems and ecology
,” Phys. Rev. E
59
(2
), 1646
(1999
). 4
C.
Grebogi
, E.
Ott
, and J. A.
Yorke
, “Critical exponent of chaotic transients in nonlinear dynamical systems
,” Phys. Rev. Lett.
57
(11
), 1284
(1986
). 5
J. A.
Yorke
and E. D.
Yorke
, “Metastable chaos: The transition to sustained chaotic behavior in the Lorenz model
,” J. Stat. Phys.
21
(3
), 263
–277
(1979
). 6
A.
Ray
, A.
Pal
, D.
Ghosh
, S. K.
Dana
, and C.
Hens
, “Mitigating long transient time in deterministic systems by resetting
,” Chaos
31
(1
), 011103
(2021
). 7
E. G.
Altmann
, J. S. E.
Portela
, and T.
Tél
, “Leaking chaotic systems
,” Rev. Mod. Phys.
85
(2
), 869
(2013
). 8
T.
Lilienkamp
, J.
Christoph
, and U.
Parlitz
, “Features of chaotic transients in excitable media governed by spiral and scroll waves
,” Phys. Rev. Lett.
119
(5
), 054101
(2017
). 9
T.
Lilienkamp
and U.
Parlitz
, “Terminal transient phase of chaotic transients
,” Phys. Rev. Lett.
120
(9
), 094101
(2018
). 10
T. M.
Lenton
, “Early warning of climate tipping points
,” Nat. Clim. Change
1
(4
), 201
–209
(2011
). 11
M.
Scheffer
, J.
Bascompte
, W. A.
Brock
, V.
Brovkin
, S. R.
Carpenter
, V.
Dakos
, H.
Held
, E. H.
Van Nes
, M.
Rietkerk
, and G.
Sugihara
, “Early-warning signals for critical transitions
,” Nature
461
(7260
), 53
–59
(2009
). 12
A.
Morozov
, K.
Abbott
, K.
Cuddington
, T.
Francis
, G.
Gellner
, A.
Hastings
, Y.-C.
Lai
, S.
Petrovskii
, K.
Scranton
, and M.
Lou Zeeman
, “Long transients in ecology: Theory and applications
,” Phys. Life Rev.
32
, 1
–40
(2020
). 13
A.
Gosztolai
, J. A.
Carrillo
, and M.
Barahona
, “Collective search with finite perception: Transient dynamics and search efficiency
,” Front. Phys.
6
, 153
(2019
). 14
R.
Martin
, M.
Schlüter
, and T.
Blenckner
, “The importance of transient social dynamics for restoring ecosystems beyond ecological tipping points
,” Proc. Natl. Acad. Sci. U.S.A.
117
(5
), 2717
–2722
(2020
). 15
J.-F.
Arnoldi
, A.
Bideault
, M.
Loreau
, and B.
Haegeman
, “How ecosystems recover from pulse perturbations: A theory of short-to long-term responses
,” J. Theor. Biol.
436
, 79
–92
(2018
). 16
J.
Gao
, B.
Barzel
, and A.-L.
Barabási
, “Universal resilience patterns in complex networks
,” Nature
530
(7590
), 307
–312
(2016
). 17
C.
Hens
, U.
Harush
, S.
Haber
, R.
Cohen
, and B.
Barzel
, “Spatiotemporal signal propagation in complex networks
,” Nat. Phys.
15
(4
), 403
–412
(2019
). 18
W.
Tarnowski
, I.
Neri
, and P.
Vivo
, “Universal transient behavior in large dynamical systems on networks
,” Phys. Rev. Res.
2
(2
), 023333
(2020
). 19
A.
Hastings
, “Transients: The key to long-term ecological understanding?
,” Trends Ecol. Evol.
19
(1
), 39
–45
(2004
). 20
A.
Hastings
, “Timescales, dynamics, and ecological understanding
,” Ecology
91
(12
), 3471
–3480
(2010
). 21
A. E.
Motter
, S. A.
Myers
, M.
Anghel
, and T.
Nishikawa
, “Spontaneous synchrony in power-grid networks
,” Nat. Phys.
9
(3
), 191
–197
(2013
). 22
L.
Halekotte
, A.
Vanselow
, and U.
Feudel
, “Transient chaos enforces uncertainty in the British power grid
,” J. Phys.: Complex.
2
(3
), 035015
(2021
).23
A.
Vanselow
, S.
Wieczorek
, and U.
Feudel
, “When very slow is too fast-collapse of a predator-prey system
,” J. Theor. Biol.
479
, 64
–72
(2019
). 24
B.
Kaszás
, U.
Feudel
, and T.
Tél
, “Tipping phenomena in typical dynamical systems subjected to parameter drift
,” Sci. Rep.
9
(1
), 1
–12
(2019
).25
U.
Feudel
, “Complex dynamics in multistable systems
,” Int. J. Bifurcation Chaos
18
(06
), 1607
–1626
(2008
). 26
B.
Nowakowski
and A. L.
Kawczyński
, “Multipeak distributions of first passage times in bistable dynamics in a model of a thermochemical system
,” ChemPhysChem: Eur. J. Chem. Phys. Phys. Chem.
7
(2
), 502
–507
(2006
). 27
H.-H.
Qiu
, Z.-J.
Yuan
, and T.-S.
Zhou
, “Distributions of the first passage time in a bistable biological system
,” Chin. J. Phys.
50
(5
), 857
–867
(2012
).28
A. L.
Kawczyński
and B.
Nowakowski
, “Stochastic transitions through unstable limit cycles in a model of bistable thermochemical system
,” Phys. Chem. Chem. Phys.
10
(2
), 289
–296
(2008
). 29
R. R.
Klevecz
, “Quantized generation time in mammalian cells as an expression of the cellular clock
,” Proc. Natl. Acad. Sci. U.S.A.
73
(11
), 4012
–4016
(1976
). 30
M.
Turcotte
, J.
Garcia-Ojalvo
, and G. M.
Süel
, “A genetic timer through noise-induced stabilization of an unstable state
,” Proc. Natl. Acad. Sci. U.S.A.
105
(41
), 15732
–15737
(2008
). 31
C.
Grebogi
, E.
Ott
, and J. A.
Yorke
, Phys. Rev. Letts.
82
, 1507
(1982
). 32
P.
Yodzis
and S.
Innes
, “Body size and consumer-resource dynamics
,” Am. Nat.
139
(6
), 1151
–1175
(1992
). 33
A.
Hastings
and T.
Powell
, “Chaos in a three-species food chain
,” Ecology
72
(3
), 896
–903
(1991
). 34
K.
McCann
and P.
Yodzis
, “Nonlinear dynamics and population disappearances
,” Am. Nat.
144
(5
), 873
–879
(1994
). 35
D.
Pattanayak
, A.
Mishra
, S. K.
Dana
, and N.
Bairagi
, “Bistability in a tri-trophic food chain model: Basin stability perspective
,” Chaos
31
(7
), 073124
(2021
). 36
M. L.
Rosenzweig
, “Exploitation in three trophic levels
,” Am. Nat.
107
(954
), 275
–294
(1973
). 37
K.
McCann
and P.
Yodzis
, “Biological conditions for chaos in a three-species food chain
,” Ecology
75
(2
), 561
–564
(1994
). 38
K.
Kempf-Leonard
, Encyclopedia of Social Measurement (Elsevier, 2004).39
C. H. C.
Little
, K. L.
Teo
, and B.
Van Brunt
, Real Analysis Via Sequences and Series
(Springer
, 2015
).© 2023 Author(s). Published under an exclusive license by AIP Publishing.
2023
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