ANALYSIS OF LONG TRANSIENTS AND DETECTION OF EARLY WARNING SIGNALS OF EXTINCTION IN A CLASS OF PREDATOR-PREY MODELS EXHIBITING BISTABLE BEHAVIOR.

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ANALYSIS OF LONG TRANSIENTS AND DETECTION OF EARLY
WARNING SIGNALS OF EXTINCTION IN A CLASS OF PREDATOR-PREY
MODELS EXHIBITING BISTABLE BEHAVIOR.
S. SADHU∗
Department of Mathematics, Georgia College & State University, GA 31061, USA.
S. CHAKRABORTY THAKUR+
Department of Physics, Auburn University, AL 36849, USA.
Abstract. In this paper, we develop a method of analyzing long transient dynamics in a class of
predator-prey models with two species of predators competing explicitly for their common prey,
where the prey evolves on a faster timescale than the predators. In a parameter regime near a
singular zero-Hopf bifurcation of the coexistence equilibrium state, we assume that the system
under study exhibits bistability between a periodic attractor that bifurcates from the singular
Hopf point and another attractor, which could be a periodic attractor or a point attractor, such
that the invariant manifolds of the coexistence equilibrium point play central roles in organizing
the dynamics. To find whether a solution that starts in a vicinity of the coexistence equilibrium
approaches the periodic attractor or the other attractor, we reduce the equations to a suitable
normal form, and examine the basin boundary near the singular Hopf point. A key component
of our study includes an analysis of the long transient dynamics, characterized by their rapid
oscillations with a slow variation in amplitude, by applying a moving average technique. We
obtain a set of necessary and sufficient conditions on the initial values of a solution near the
coexistence equilibrium to determine whether it lies in the basin of attraction of the periodic
attractor. As a result of our analysis, we devise a method of identifying early warning signals,
significantly in advance, of a future crisis that could lead to extinction of one of the predators.
The analysis is applied to the predator-prey model considered in [Discrete and Continuous
Dynamical Systems - B 2021, 26(10), pp. 5251-5279] and we find that our theory is in good
agreement with the numerical simulations carried out for this model.
Keywords. Long transients, method of averaging, zero-Hopf, singular Hopf, bistability, early
warning signals, slow-fast systems, predator-prey models.
AMS subject classifications. 34C20, 34C29, 34D15, 37C70, 37G05, 37G35, 92D40.
1. Introduction
Identifying early warning signals for anticipating population collapses or finding early indicators
of recovery of a threatened or an endangered population is a major focus of research in nature
conservation and ecosystem management. Often the collapses can be catastrophic as they may not
be reverted, causing huge changes in ecosystem structure and function while leading to extinction
of species and substantial loss of biodiversity [19, 20, 28]. Examples of catastrophic collapses
include the extinction of Steller’s sea cow Hydrodamalis gigas within just 30 years of its discovery
in the late 18th century, the dramatic extinction of passenger pigeon Estopistes migratorius in
the early 20th century, the collapse of the Atlantic northwest cod fishery in the early 1990s, the
extinction of sea urchins Paracentrotus lividus in South Basin of Lough Hyne in Ireland in the
early 2000s, and sharp decline of coral cover on the Great Barrier Reef. While collapse of a
E-mail addresses:susmita.sadhu@gcsu.edu∗, szc0199@auburn.edu+.
1
arXiv:2210.04097v2 [math.DS] 27 Apr 2024
2 BISTABLE DYNAMICS AND EARLY WARNING SIGNALS
population is of primary concern for conservationists, early indicators of recovery of a system can
play a crucial role in assessing the effectiveness of intervention strategies and resource management
decisions [11]. Consequently, changes in abundance or fitness-related traits are usually monitored
to find indicators of impending shifts in population dynamics of threatened species. A common
trend observed in the population of many endangered species is an abrupt transition from a
seemingly steady state, persisting over many generations, to an unsustainable state that may
lead to its extinction. The persistence of long transient phases in population ecology has been
also documented in many empirical observations and theoretical models of various species [19,
20], which then brings forth the subject of developing suitable mathematical methods to analyze
transient dynamics in ecological models.
Abrupt shifts in ecosystems or “regime shifts” have been attributed to bifurcations in the under-
lying system driven by slow changes in environmental parameters, random fluctuations or external
forcing, the drifting rate of time-dependent parameters, or a combination of some of these factors
[1, 38, 39, 41]. Early warning signals of regime shifts have usually been related to the phenomenon
of “critical slowing down” [8] that arises in the vicinity of local bifurcations, and the commonly
known measures for detecting them include examination of changes in trends of statistical mea-
sures such as variance, autocorrelation, and skewness in time series [6, 38, 39]. However, there
is also a growing recognition of the leading role of long transient dynamics in explaining regime
shifts and developing strategies for efficient ecological forecasting, since regime shifts or critical
transitions between alternative states can be viewed as phenomena induced by transients-related
mechanisms [20, 28]. In this paper, we take the latter outlook to study regime shifts. More-
over, rather than considering statistical indicators of critical transitions, we take an alternative
approach for detecting early warning signals by identifying and monitoring a suitable combination
of the state variables that resulted from analyzing the mechanism driving the long transients in
the system. Though the importance of long transients has been recognized in mathematical ecol-
ogy [19], analytical techniques for studying transient dynamics in relevant timescales in three or
higher-dimensional models is still at its mathematical infancy [22, 36]. With this spirit, in this
paper, we adopt a dynamical systems approach to analyze long transients and apply the analysis
to predict an impending transition in population dynamics of a class of three-species predator-prey
models exhibiting bistability between a limit cycle and a boundary equilibrium state in a param-
eter regime near a singular zero-Hopf bifurcation [2, 4, 15, 17, 18, 27]. The goal is to predict the
long-term behaviors of solutions exhibiting similar oscillatory dynamics as transients and devise a
method of identifying an early warning signal significantly in advance of an abrupt transition that
could lead to an extinction.
The work in this paper is inspired by the dynamics of the model studied in [35] which reads as
dR
dT =rR 1−R
K−p1RC1
H1+R−p2RC2
H2+R
dC1
dT =q1p1RC1
H1+R−m1C1−a12C1C2
dC2
dT =q2p2RC2
H2+R−m2C2−a21C1C2−a22C2
2,
(1)
where Rrepresents the population density of the prey; C1,C2represent the densities of the two
species of competing predators; r > 0 and K > 0 are the intrinsic growth rate and the carrying
capacity of the prey; p1>0 is the maximum per-capita predation rate of C1,H1>0 is the semi-
saturation constant which represents the prey density at which C1reaches half of its maximum
predation rate (p1/2), q1>0 is the birth-to-consumption ratio of C1,m1>0 is the per-capita
natural death rate of C1, and a12 >0 is the rate of adverse effect of C2on C1. The other
parameters p2, q2, m2, H2, a21 are defined analogously and a22 >0 represents the intraspecific
competition within C2. It is assumed that Revolves on a faster timescale than C1and C2, and
C2is a stronger competitor than C1. Depending on the relative strengths of a22 and a21,C2
may either outcompete C1, or the three species may coexist in an equilibrium or an oscillatory
state in the form of small-amplitude oscillations, mixed-mode oscillations [7, 15, 34], or relaxation
oscillations.
BISTABLE DYNAMICS AND EARLY WARNING SIGNALS 3
System (1) in its non-dimensional form (see [35] for the details) can be written as
ζ˙x=x1−x−y
β1+x−z
β2+x:= xϕ(x, y, z)
˙y=yx
β1+x−d1−α12z:= yχ(x, y, z)
˙z=zx
β2+x−d2−α21y−hz:= zψ(x, y, z),
(2)
where the overdots denote differentiation with respect to the slow time s=ζrT , where ζmeasures
the ratio of the growth rates of the predators to the prey and satisfies 0 < ζ ≪1. The nontrivial
x,y, and z-nullcline in (2) are denoted by ϕ= 0, χ= 0, and ψ= 0 respectively. The dimensionless
parameters β1,d1and α12 respectively represent the predation efficiency [13], rescaled mortality
rate and rescaled interspecific competition coefficient of y. The parameters β2,d2, and α21 are
analogously defined and hrepresents the rescaled intraspecific competition coefficient within z.
Interestingly, it was observed in [35] that for suitable parameter values near a subcritical singular
Hopf bifurcation, system (2) exhibits bistability between a small-amplitude limit cycle and a
boundary equilibrium state which corresponds to the extinction of the weaker predator (figure 21
in [35]). It turns out that in this regime, the dynamics are organized by the invariant manifolds
of a nearby saddle-focus coexistence equilibrium point, allowing a solution that starts near the
coexistence equilibrium point to approach it along its two-dimensional stable manifold and then
escape along its one-dimensional unstable manifold either towards the boundary equilibrium state
or towards a nearby limit cycle (see figure 2). In both cases, the local dynamics near the coexistence
equilibrium appear very similar and are characterized by high-frequency oscillations with slowly
varying amplitude along the invariant manifolds of the equilibrium. The similarity in the initial
trends makes identification of early warning signals of a population collapse extremely challenging
(see figures 1 and 2), which is one of the subjects of investigation in this work.
0 100 200 300 400
s
0
0.2
0.4
0.6
0.8 x
y
z
0 10 20 30
0.25
0.3
0.35
0.4
0.45
a1
a2
a1
0 10 20 30
0.116
0.117
0.118
0.119
a2
(a) IC: (0.2785,0.1181,0.4164)
0 100 200 300 400
s
0
0.2
0.4
0.6
0.8 x
y
z
0 10 20 30
0.25
0.3
0.35
0.4
0.45
0 10 20 30
0.116
0.117
0.118
0.119
b1
b1
b2
b2
Population
collapse
(b) IC: (0.2831,0.1184,0.417)
Figure 1. An illustration of bistable behavior exhibited by system (2) for h=
0.2649 and other parameter values as in (5)). Both solutions exhibit similar
patterns of oscillations initially (see insets a1-a2 and b1-b2) but have different
asymptotic behaviors. IC: initial conditions.
Another aspect that is considered in this study pertains to an analysis of the mechanism un-
derlying long transients in systems exhibiting such bistable dynamics. Some of the commonly
known mechanisms responsible for the emergence of long transients in ecological models include
the proximity of a system to a ghost attractor, slow passage through a saddle known as a “saddle
crawl-by”, presence of slow manifolds in systems with timescale separation, or effects of stochas-
ticity or time delays [20, 28]. However, in this paper, we reveal another mechanism that can
lead to long transients, namely, a slow passage through a saddle-focus equilibrium that lies near
a zero-Hopf bifurcation in the parameter space. It turns out that the proximity of the system
to a zero-Hopf bifurcation results in a weak contraction and a weak expansion along the stable
4 BISTABLE DYNAMICS AND EARLY WARNING SIGNALS
(a) IC: (0.2785,0.1181,0.4164) (b) IC: (0.2831,0.1184,0.417)
Figure 2. The phase portraits of the transient dynamics corresponding to the
time series in figure 1 for s∈[0,100]. The solutions spend a significant amount
of time near the coexistence equilibrium point shown by the red dot before ap-
proaching their asymptotic states. The initial conditions are shown by the blue
dots. (A) The solution asymptotically approaches a periodic attractor. (B) The
solution asymptotically approaches a boundary equilibrium point.
and unstable manifolds of the saddle-focus equilibrium, allowing a solution to spend a significant
amount of time near the invariant manifolds as it passes through the equilibrium, thus generating
long transients as shown in figure 2. To the best of our knowledge, such a mechanism underlying
long transients in three or higher-dimensional models has not been explored yet.
To analyze the bistable behavior and the transient dynamics, the general class of models with
two intrinsic timescales, which includes system (1) as a special case, is reduced to a topologically
equivalent class, referred to as the normal form [4], which is valid near the singular Hopf point
where certain conditions on the derivatives hold. The normal form consists of two-fast variables
and one-slow variable, where the slow variable represents the slowly changing dynamics along the
axial direction corresponding to the unstable set of the saddle-focus equilibrium, while the fast
variables feature rapid oscillations with slow variation in amplitude that occur along the stable set
of the equilibrium. Consequently, the transformed coordinates system associated with the normal
form naturally captures the “near cylindrical symmetry” and the intrinsic nature of the dynamics
near the saddle-focus equilibrium. Exploiting the separation of timescales between the rapid
oscillations and the slow variation in the amplitude, we apply the method of averaging [18, 40]
to partition the system into the sum of its time-averaged and fluctuating parts and consider the
equation of the time evolution of the mean of the oscillations in the axial direction. We solve
the associated ODE and express the solution in terms of the mean of the oscillations of the fast
subsystem, and study its properties. The analysis is then used to find a set of sufficient conditions
to distinguish between solutions that start in a vicinity of the saddle-focus point with contrasting
long-term behaviors.
The normal form also reveals the underlying geometrical structure of the system and gives
insight into the landscape of the basin boundary of the two attractors in the vicinity of the saddle-
focus equilibrium. We explicitly find a region in the phase space, bounded by the surface of an
elliptic paraboloid, that lies in the basin of attraction of the periodic attractor. The 3-D region,
which we refer to as the “funnel” encloses a branch of the unstable manifold of the saddle-focus
equilibrium and separates orbits approaching the limit cycle from the point attractor. We prove
that a solution in the vicinity of the stable manifold of the saddle-focus equilibrium must enter
into this “funnel” to asymptotically approach the periodic attractor. We find a set of necessary
and sufficient conditions on the initial data corresponding to which a solution eventually enters the
funnel as it escapes along the unstable manifold of the saddle-focus equilibrium. Consequently, we
BISTABLE DYNAMICS AND EARLY WARNING SIGNALS 5
obtain a critical threshold on the slowly varying dynamics along the unstable manifold, such that
if a solution goes below it, it must approach the boundary equilibrium state. Finally, the analysis
is used to devise a method for finding early warning signals of the onset of a drastic change in
the population of one of the species of predators. The results are in good agreement with the
numerical simulations carried out for system (1).
The remainder of the paper is organized as follows. In Section 2, we present the general formu-
lation of the class of equations and lay down the assumptions. We briefly discuss the geometric
structure of system (1) and perform few numerical investigations to gain insight into the dynamics.
In Section 3, we reduce the general system to a topologically equivalent form, which is valid near
the singular Hopf bifurcation and analyze the equivalent system. We obtain a set of sufficient
conditions on the initial data in a neighborhood of the coexistence equilibrium to lie in the basin
of attraction of the periodic attractor. Using the analysis, we then devise a method of detecting an
early warning signal of population collapse in Section 4. The results are supported by numerical
simulations in Section 5. Finally, we summarize our conclusion in Section 6.
2. Mathematical Model
2.1. General Formulation. The class of equations under consideration is of the form
ζ˙x=f1(x, y, z, h) := xϕ(x, y, z, h)
˙y=f2(x, y, z, h) := yχ(x, y, z, h)
˙z=f3(x, y, z, h) := zψ(x, y, z, h),
(3)
where x, y, z rehresent the population densities of the prey and the two species of predators respec-
tively, pis a parameter in a compact subset of Rand ϕ, χ, and ψare smooth functions having the
properties described by (H1)-(H7) and (Q1)-(Q5) below. The overdots in (3) denote differentiation
with respect to the time variable sand 0 < ζ ≪1 represents a singular parameter. We assume
that the prey evolves on a faster timescale than the predators, a phenomenon commonly observed
in many ecosystems [13, 14, 29, 32]. We make the following assumptions:
(H1) ϕ(0,0,0, h)>0, χ(0,0,0, h)<0, ψ(0,0,0, h)<0, ϕ(1,0,0, h) = 0, χ(1,0,0, h)>0,
ψ(1,0,0, h)>0 and ϕx(1,0,0, h)<0.
(H2) ϕx(x, y, z, h) = 0 defines a smooth curve Fon the surface S:= {(x, y, z)∈R3:
ϕ(x, y, z, h)=0}dividing Sinto two smooth surfaces:
Sa={(x, y, z)∈S:ϕx(x, y, z, h)<0}, Sr={(x, y, z)∈S:ϕx(x, y, z, h)>0}.
Condition (H1) implies that the equilibrium points (0,0,0) and (1,0,0) are saddles, where
(0,0,0) is attracting in the invariant yz-plane and repelling along the invariant x-axis while (1,0,0)
is attracting along the invariant x-axis and repelling along the yz directions. Depending on h,
(3) may admit other equilibria, some of which may lie on the xy-plane or the xz-plane. These
equilibria will be referred to as the boundary equilibria and will be denoted by Exy and Exz
respectively. Rescaling sby ζand letting t=s/ζ, system (3) can be reformulated as
x′=f1(x, y, z, h)
y′=ζf2(x, y, z, h)
z′=ζf3(x, y, z, h),
(4)
where the primes denote differentiation with respect to t. System (4) is referred to as the fast
system, whereas (3) as the slow system. The set of equilibria of (4) in its singular limit is called
the critical manifold [15], M:= Π ∪S, where Π is the invariant yz plane and Sis the surface as
defined in (H2). Hypothesis (H2) implies that the surface Sais normally attracting while Sris
normally repelling with respect to the limiting fast system. The normal hyperbolicity of Sis lost
via saddle-node bifurcations along the fold curve F.
We assume that there exists a point (¯x, ¯y, ¯z, ¯
h) such that the linearization of (3) at this point
has a pair of eigenvalues that approach infinity as ζ→0 and the following conditions on f1,f2,
f3and their derivatives hold at (¯x, ¯y, ¯z, ¯
h):
摘要：

ANALYSISOFLONGTRANSIENTSANDDETECTIONOFEARLYWARNINGSIGNALSOFEXTINCTIONINACLASSOFPREDATOR-PREYMODELSEXHIBITINGBISTABLEBEHAVIOR.S.SADHU∗DepartmentofMathematics,GeorgiaCollege&StateUniversity,GA31061,USA.S.CHAKRABORTYTHAKUR+DepartmentofPhysics,AuburnUniversity,AL36849,USA.Abstract.Inthispaper,wedevelopa...

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