# Multimodal distribution of transient time of predator extinction in a three-
species food chain

Debarghya Pattanayak1, Arindam Mishra2, Syamal K. Dana1,2,3, Nandadulal
Bairagi1 1 Centre for Mathematical Biology and Ecology, Department of
Mathematics, Jadavpur University, Kolkata 700032, India.
2 Division of Dynamics, Lodz University of Technology, Stefanowskiego 1/15,
90-924 Lodz, Poland
3 Department of Mathematics, National Institute of Technology, Durgapur
713209, India.

###### Abstract

Studies of transient dynamics captures the time history of any dramatic
changes in the dynamics of a system. The transient dynamics is investigated
here in a classic ecological model of a bistable tri-trophic food chain. All
the species in the food chain either coexist or undergoes a partial extinction
with the loss of predator after a transient time that depends upon the initial
density of species populations. The distribution of transient time of
extinction of the predator in response to initial states shows interesting
pattern of inhomogeneity and anisotropy in the basin of predator-fee state.
Precisely, the distribution shows a multimodal character when the initial
points are near the basin boundary, and unimodal at locations far from the
border. The distribution is also anisotropic in the sense that the number of
modes depends on the direction from the basin location. We define two new
metrics, viz., homogeneity index and local isotropic index to characterize the
distinctive features of the distribution. The inhomogeneity of distribution
reduces significantly with resource enrichment, however, a similar change in
anisotropy is relatively low. We try to explain the origin of such multimodal
distributions from the dynamical system perspectives and present its
ecological implications.

††preprint: APS/123-QED††thanks: Corresponding author:
<EMAIL_ADDRESS>
##  I Introduction

A study of asymptotic behavior gives a partial picture of dynamics of a system
[1, 2]. The behavior of a system for the intermediate time remains unknown
before it reaches the asymptotic state. In particular, in ecological systems,
a study of only the asymptotic behavior, therefore, may hinder any attempt to
exercise any form of control on the system’s behavior at its initial stage,
that shall beneficially affect its long-term behavior, such as inducing any
support for survival to the vulnerable species in the transient state, which
may lead to the coexistence of all the species in the ecosystem [3]. Moreover,
ignoring the underlying dynamics that a system may exhibit from its initial
point to its long-term behavior keeps us oblivious to the nature of the
evolution of the system. The dynamics that exists before reaching an
asymptotic state is called the transient dynamics, and the time required for
any trajectory of a system to reach the final attractor (asymptotic state)
from the initial state is called the transient time [4, 5, 6]. In recent
years, the statistics analysis of transient time has received a considerable
attention due to its relevance in complex systems [7, 8, 9], climatic models
[10, 11], ecology [12, 13, 14, 15, 16] and dynamical networks [17, 18], in
general. The statistics of transient time is important for predicting extreme
events [19], species extinction [3, 20, 21, 1] and controlling power grid
failure [22]. The role of transient time becomes all the more profound for
assessing the critical transitions to alternate stable states or predicting a
tipping point [23].

The statistics of transient time is more complicated for multistable systems
[24], where the choice of initial states affects the transient dynamics.
Hence, the statistics of transient time needs to be assessed against the
pertaining sub-basins [25] of the multistable system. The challenges posed to
such analysis become all the more aggravated when multimodal distribution of
transient time arises, making our conventional statistical approaches of mean,
median, and other central tendencies difficult to apply. In the event of
multiple modes in the distribution, neither following any known patterns,
normal, Poisson, nor the modes being equispaced, the only choice left with us
is the analysis of each distinct mode in the multimodal distribution.
Moreover, when dealing with such multistable systems displaying multimodal
distributions, the observation of the variation in the number of distinct
modes, along with other statistical parameters, simulated with initial points
at different distances and on different directions of the basin boundary,
become a matter of utmost concern. If any noticeable trends are found in the
obtained results, attempts could be made to characterize the structure of the
basin from the perspective of transient time distributions.

We address the problem of analyzing the multimodal distribution of transient
time that appears in a paradigmatic tri-trophic food chain model [26, 27, 28],
which is well known for its bistability. The system either shows coexistence
of all the species or a predator-free limit cycle depending on the initial
population size and parameter values. What has not been explored so far, in
this bistable system, is the distribution of transient time of the predator’s
extinction. The focal point of our study is to examine the distribution of
this transient time that appears to be multimodal, and to provide some
techniques for its analysis. We adopt a method of partitioning the concerned
basin of attraction into identical blocks of reasonably small size and then
analyze the transient time and its distributions in each block, separately,
and subsequently build a larger picture of the entire distribution scenario of
the system. During this process, efforts are also given to characterize the
basin structure of the model system from the perspective of distribution of
the transient time, which appears to be inhomogeneous and anisotropic. Broadly
speaking, the number of modes varies with the distance of the initial states
from the basin boundary and also depends upon the direction of the initial
points relative to the same boundary.

The organization of the paper is as follows. A brief introduction of the model
and its bistability with the basin of attraction are presented in Section II.
Section II.1 discusses the trends in the distribution of transient time of
extinction of the predator and the reason behind the existence of such
multimodal distributions. In Section III, a description is given how the basin
of predator-free limit cycle is partitioned into small identical blocks and
the multimodal distribution is derived. Two new metrics are defined to
quantify the structural pattern of the basin with respect to the number of
modes in the transient time distributions. Subsequently, attributes of the
distributions in terms of the new metrics are discussed to explain the noted
multimodal distribution pattern. The paper ends with a brief discussion in
Section IV.

##  II Tri-tropic food chain model

The tri-trophic food chain [29, 27, 26] presents the interactions between a
resource, a consumer and a predator. The model assumes that the resource grows
in a logistic fashion in the absence of any consumer, whereas the consumer and
predator both exponentially decay in the absence of their respective prey. The
consumer and predator have saturated Holling Type-II functional responses, and
there are no losses of time and energy during hunting and digestion. Three
species grow simultaneously and have overlapping generations with no age
structure, where the growth rates of resource population density ($R$),
consumer population density ($C$) and predator population density ($P$) at
time $t$ are defined by

$\begin{split}\frac{dR}{dt}&=R(1-\frac{R}{K})-x_{C}y_{C}\frac{CR}{R+R_{0}},\\\
\frac{dC}{dt}&=-x_{C}C(1-y_{C}\frac{R}{R+R_{0}})-x_{P}y_{P}\frac{PC}{C+C_{0}},\\\
\frac{dP}{dt}&=-x_{P}P(1-y_{P}\frac{C}{C+C_{0}}).\end{split}$ (1)

,where $R(0),C(0),P(0)>0$.
The default parameter values as used for all the simulations here, in this
study, are taken from McCann and Yodzis [30] and shown in Table 1. The
parameter values are biologically plausible with a feasible prey-predator body
mass ratio and consistent with the consumer or predator being a vertebrate or
invertebrate ectotherm [30].

Table 1: Parameters description and their default values [30]. Parameter |  Description |  Value
---|---|---
$K$ |  Resource carrying capacity |  0.95
$R_{0}$ |  Resource half-saturation density |  0.161
$C_{0}$ |  Consumer half-saturation density |  0.5
$x_{C}$ |  Energy consumption rate of $C$ per unit body weight relative to the rapidity with which the resource replaces itself |  0.4
$x_{P}$ |  Energy consumption rate of $P$ per unit body weight relative to the rapidity with which the resource replaces itself |  0.08
$y_{C}$ |  Energy spent per unit energy acquired by $C$ |  2.01
$y_{P}$ |  Energy spent per unit energy acquired by $P$ |  2.876

This classic food chain model has been elaborately investigated as reported
earlier [31, 28] and it is bistable for a broad range of parameters where two
dynamical regimes exist with their distinct basins separated by a smooth
boundary. If the initial population densities of species are taken from one of
the basins, they lead to a coexisting attractor [28] depending upon the choice
of parameters. For an alternative choice of initial states from the other
basin, the trajectories will enter the predator-free period-1 limit cycle. A
prototype illustration of bistability and transient dynamics of predator
extinction as typically seen in the model is presented in Fig. 1 for a
selective choice of parameters.

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(a)

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(b)

Figure 1: Bistability and transient dynamics of the 3-species food chain. (a)
A chaotic attractor (blue) coexists with a predator-free period-1 limit cycle
(red line) in $R-C-P$ state space. (b) Transient dynamics of the predator
population is shown with its extinction when the system transits to an
asymptotic state of limit cycle oscillation ($P=0$).

###  II.1 Distribution of transient time

For a broad range of parameters, in the initial stages, the predator coexists
in chaotic or periodic oscillations with resource and consumer and then
experiences an extinction as shown in in Fig. 1(b). Fig. 2(a) shows the
bistable basin, where the white basin corresponds to coexistence dynamics
(periodic for this parameter set) of the three pertaining species and the blue
basin represents the period-1 limit cycle when the predator goes to extinction
after a transient time. We compute the distribution of transient time of
predator extinction and search for its possible variation with the location of
initial points. For this purpose, we make an preliminary check with a
selection of four adjacent identical square blocks (green, magenta, black and
red) of edge $10^{-1}$ units in the blue basin shown in Fig. 2(a). The square
blocks are arranged horizontally in a sequence of increasing distance from the
basin boundary. Subsequently, a large number of initial points ($10^{5}$) are
chosen randomly from each of the colored blocks and the transient time of the
trajectory for all the selected initial points in each block is numerically
estimated and their distribution is plotted. This is repeated for all the four
colored boxes and plotted separately, when a distinct characteristic feature
is obtained with a variation in the distribution of transient time in the
blocks. A multimodal distribution of transients is observed and the number of
modes varies for each block, along with a variation in the mean transient
time. Figs. 2(b)-(e) present a picture of a decreasing number of modes in the
distribution as we move from one block to another, away from the basin
boundary to make our choices of initial points. This distinctive qualitative
feature of distributions and the trend remains invariant with the number of
initial points (checked with different number of points), provided they are
chosen randomly and sufficiently large in numbers.

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(a)

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(b)

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(c)

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(d)

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(e)

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(f)

Figure 2: Basin of attraction of the bistable tri-tropic food chain and
distribution of transient time. (a) Basin of attraction in a $C-P$ plane with
two distinct basins separated by a smooth boundary. White basin for coexisting
species, and blue basin for predator-free limit cycle. Location of the green
box ($[0.6,0.7]\times[0.8,0.9]$), magenta ($[0.7,0.8]\times[0.8,0.9]$), black
($[0.8,0.9]\times[0.8,0.9]$) and red ($[0.9,1.0]\times[0.8,0.9]$) boxes in the
blue basin. Modes of distribution of transient time: (b) trimodal, (c)
bimodal, (d) bimodal, and (e) unimodal. Red curves in (b, c, d, e) are the
fitted lines of distributions, which are log-normal. (f) Mean transient time
($MTT$) calculated for multiple modes in each box, decreases with distance of
the boxes from the basin boundary. The red line is the regression line.
Parameter values are given in Table 1.

Figure 2(b) is produced with initial points chosen from the block (green) at a
grid location $[0.6,0.7]\times[0.8,0.9]$ closest to the basin boundary when
three distinct modes are observed. As we move away from the basin boundary to
the adjacent blocks (magenta and black) at grid locations
$[0.7,0.8]\times[0.8,0.9]$ (magenta) and $[0.8,0.9]\times[0.8,0.9]$ (black),
the distribution changes from tri-modal to bimodal as shown in Fig. 2(c) and
Fig. 2(d), respectively. One mode for initial conditions from the black box is
not so prominent, yet the distribution is clearly bi-modal, indicating the
decreasing number of modes with the increasing distance from the basin
boundary. The furthest block (red) at location $[0.9,1.0]\times[0.8,0.9]$
shows unimodal distribution of transient time of predator extinction in Fig.
2(e). The mean transient time ($MTT$) is plotted for each block in Fig. 2(f),
and interestingly, it also decreases with the increasing distance of the boxes
from the basin boundary. Focusing separately on each mode for all the blocks
in Figs. 2(b)-(e), we opine that each mode follows a log-normal distribution
(fitted with red lines). The statistical properties such as mean, variance and
deviations (fluctuations) in transient time for each block are presented in
Table 2.

###  II.2 Origin of multimodal distribution

We address an obvious question how the distinct modal distributions of
transient time originate. Figures 3(a)-(d) show exemplary trajectories of the
3-species food chain in $R-C-P$ space for different choices of initial states,
taken from the middle of the four colored blocks in Fig. 2(a). The qualitative
picture of a trajectory remains unchanged for varying choices of initial
conditions within each box. It reveals that the system follows different
pathways to arrive at a predator-free state, through spiraling for a longer
run and alternatively, via grazing the plane $C=0.01$ (yellow-colored plane,
as evident from our data) followed by a shorter run on spiraling. For
illustrations, we consider four particular initial values, taken from each
colored box, with only a change in the initial consumer density. For the
initial state from the green box $(0.5,0.65,0.85)$, the trajectory spirally
descends to reach the predator-free state ($P=0$) avoiding the yellow plane at
$C=0.01$. For another set of initial points from the magenta box
$(0.5,0.75,0.85)$, the trajectory comes closer to the plane $C=0.01$, in the
beginning, yet not touching it as shown in Fig. 3(b), before finally spiraling
on to the $P=0$ plane of the stable limit cycle. The trajectories shown in
Figs. 3(c-d) are somewhat similar in nature with initial conditions taken from
the black box $(0.5,0.85,0.85)$ and the red box $(0.5,0.95,0.85)$,
respectively. In both the cases, the trajectory first grazes the $C=0.01$
plane for some duration of time and then make a short run of spiraling on
before reaching the $P=0$ plane of the limit cycle.

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(a)

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(b)

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(c)

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(d)

Figure 3: Trajectories of the tri-tropic food chain model. 3-dimensional
trajectories for a variety of choices of initial points from separate blocks
showing different modes. Transient dynamics of the system when initial points
are chosen from (a) green box at (0.5,0.65,0.85) in Fig. 2(b), (b) magenta box
at (0.5,0.75,0.85) in Fig. 2(c), (c) black box at (0.5,0.85,0.85) in Fig. 2(d)
and (d) (0.5,0.95,0.85) from the red box in Fig. 2(e), respectively. Exemplary
trajectories for different choices of initial points show a decay of the
predator population while grazing the light yellow plane at $C=0.01$, and then
make a descent to the $P=0$ plane with a spiraling motion. Parameters are
given in Table 1.

We denote the total transient time of any trajectory as $T$, counting from the
initial state and the duration of reaching the predator-free limit cycle
state. The entire transient time $T$ is divided into three time segments when
the trajectory is assumed to spend (i) $T_{1}>0$ time to reach a close
location of the ($C=0.01$, yellow) plane from the initial state, (ii)
$T_{2}\geq 0$ time spent on ($C=0.01$, yellow) plane, (iii) $T_{3}>0$ time of
spiraling before reaching the predator-free limit cycle. The total time $T$
taken by a trajectory to arrive at the predator-free limit cycle from any
initial states is

$\begin{split}T=T_{1}+T_{2}+T_{3}.\end{split}$ (2)

Figures 3 show that $T_{1},T_{3}\neq 0$, for all the choices of initial
states, however, $T_{2}$=0 when the trajectory never grazes the $C=0.01$
plane. When the trajectories descent in such a pure spiral motion, avoiding
the $C=0.01$ plane, the radius of evolution of a spiral gradually increases in
each cycle before, finally, arriving at the asymptotic state of stable
predator-free limit cycle. The average time required to complete one
revolution while spiraling is assumed to be $T^{{}^{\prime}}$.

For the first case in Fig. 3(a), we evaluate $T_{1}=24.79$, $T_{2}=0$ (since
the trajectory avoids the $C=0.01$ plane never grazing it) and
$T^{{}^{\prime}}=27.7$. We use a Poincaré surface of section to estimate the
time intervals $T_{1}$ and $T^{\prime}$ (see Appendix). If $n\in Z_{+}$
denotes the total number of revolutions during spiraling before reaching the
predator-free limit cycle, we obtain from Eq. (2)

$\begin{split}T=24.79+27.7n.\end{split}$ (3)

For the trimodal distribution in Fig. 2(b), we estimated the mean transient
time $MT_{1},MT_{2},MT_{3}$ for each of the three distinct modes,

$\displaystyle MT_{1}$ $\displaystyle=$ $\displaystyle
467.88=24.79+443.09\approx+T_{1}+16\times T^{{}^{\prime}},$ $\displaystyle
MT_{2}$ $\displaystyle=$ $\displaystyle 440.57=24.79+415.78\approx
T_{1}+15\times T^{{}^{\prime}},$ $\displaystyle MT_{3}$ $\displaystyle=$
$\displaystyle 413.08=24.79+388.29\approx T_{1}+14\times T^{{}^{\prime}}.$

The above expressions support a fact that the variation in total transient
time $T$ mainly depends on the number of revolutions during spiraling, which
appears as an integral multiple of $T^{\prime}(n=14,15,16)$ while $T_{1}$
remains constant ($24.79$) for all the three modes (see Appendix for details).
This difference in the number of revolutions leads to three distinct modes of
distribution.

For the trajectory in Fig. 3(d), $T_{2}\neq 0$, and we find

$T_{1}=8.39,~{}T_{2}=22.61~{}\mbox{and}~{}T^{{}^{\prime}}=27.65.$

For the unimodal distribution in Fig. 2(e), we estimate the mean transient
time for the unique mode as $MT=366.47$ following Eq. (2),

$MT=366.47=8.39+22.61+335.47\approx T_{1}+T_{2}+12\times T^{{}^{\prime}}.$

As we move away from the basin boundary and make a choice of initial points
from the subsequent blocks, we see that the trajectories come closer to the
$C=0.01$ plane, and in some cases, the trajectory even grazes the plane and
spend more time on it as shown in Figs. 3(b-d). As a result, $T_{1}$ and
$T_{2}$ become larger, and the trajectories spend lesser time during
spiraling. This is reflected in the lesser number of revolutions undertaken by
the trajectory before reaching the predator-free state, and hence less number
of modes are seen. Furthermore, in Fig. 3(d), the trajectory takes a large
amount of time on the $C=0.01$ plane and shows very few revolutions during
spiraling, rather quickly arriving at the predator-free limit cycle.

The fact of decreasing mean transient time ($MTT$) with distance from the
basin boundary as shown in Fig. 2(f), is also explained from Figs. 3(a-d). As
the initial point is chosen from a location far away from the basin boundary,
a larger transient time is spent by the trajectories on the $C=0.01$ plane.
Since this plane makes the trajectories approach the predator-free period-1
limit cycle faster than approaching it through spiraling motion in the space
without touching the vertical yellow plane, the mean transient time decreases
as the initial points are chosen further away from the basin boundary. The
transient time and standard deviation (fluctuations) for different modes for
all the colored boxes are presented in Table 2.

Table 2: Statistical measures Color | Mode Sl. no. | MTT | MTT (all modes) | Std. deviation
---|---|---|---|---
Green | 1 | 413.08 |  | 2.07
| 2 | 440.57 | 440.51 | 3.08
| 3 | 467.88 |  | 1.39
Magenta | 1 | 390.75 | 401.15 | 1.31
| 2 | 411.54 |  | 0.72
Black | 1 | 370.58 | 379.34 | 0.52
| 2 | 388.10 |  | 0.67
Red | 1 | 366.47 | 366.47 | 0.97

##  III Basin Structure: Distribution pattern of transients

In order to find a broader picture of the distribution pattern of transient
time throughout the basin, we divide the entire blue region (Fig. 2a) into
identical square blocks $N_{i},i\in\mathbb{Z_{+}}$, of edge length $10^{-1}$
units such that $N_{i}=\\{(x,y)\in\mathbb{R_{+}}^{2}:0.1(n-1)<x\leq
0.1n,~{}0.1(m-1)<y\leq 0.1m,~{}~{}n,m=1,2,....,10\\}.$ From each $N_{i}$, a
number of $10^{5}$ random initial points are chosen, and the distribution of
transient time of the trajectories is mapped in Fig. 4(a), where the number of
modes in a square block $N_{i}$, $M(N_{i})$, is represented by the color bar.
Keeping at par with the previous section, the number of modes is found larger
in the region near the boundary between the white and blue basins. As we move
away from the basin boundary, the number of modes decreases and becomes
minimal at the farthest location from the basin boundary. The basin clearly
shows fluctuation in the number of modes that vary with the location of
initial points. More categorically, as shown in Fig. 4(b), a 5-mode
distribution is seen on the left-hand side of the white basin. The bottom and
the right-hand side of the boundary displays $4,~{}3,~{}2$-mode distribution
with a unimodal distribution at top right corner of the basin. The basin is
mainly dominated by bimodal distributions (dark blue), with traces of unimodal
distributions existing at the upper right corner. Besides such pattern in the
distribution of modes, directional variation in the number of modes exists,
indicating an anisotropy in the distribution pattern of transients in the
basin. We quantify such inhomogeneity and anisotropy of the basin structure by
introducing two new metrics.

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(a)

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(b)

Figure 4: Pattern of distribution of transients in the basin of the tri-tropic
food chain model. (a) Various color shades in the basin correspond to the
number of modes in the distribution of transient time for each $N_{i}$. The
color bar denotes the number of modes. (b) Numbers in each block mark the
number of modes in the distribution of transient time. Three lines (yellow,
green, and red) represent the shortest distance from the basin boundary to the
points ($Q_{1},Q_{2},Q_{3}$), where a homogeneity index is calculated. Yellow
circles mark the boxes through which the lines pass.

###  III.1 Inhomogeneity of distribution pattern of transients in the basin

The very presence of multimodal distributions in the transient time, obtained
from initial points in the blue basin, implies inhomogeneity of the basin
topology [32]. To quantify this property, we define a homogeneity index ($HI$)
of any point $Q$ in the basin. We consider the shortest distance, $L$,
measured from the point $Q$ to the boundary of the white basin, and calculate
the reciprocal of the mean of the number of modes, $\\{M(N_{i})\\}$, in all
the square blocks, $\\{N_{i}\\}$, through which the $L$ line passes. Thus the
homogeneity index of the point $Q$, $HI(Q)$, is given by

$HI(Q)=\left[\frac{\Sigma(M(N_{i}))}{\mathbf{card}(N_{i})}\right]^{-1}$

, where $\mathbf{card}(N_{i})$ is the number of boxes through which the line
$L$ passes and $M(N_{i})$ is the number of modes in the block $N_{i}$.
For illustration, we consider three points $Q_{1},\,Q_{2},\,Q_{3}$ in the blue
basin in Fig. 4(b). The shortest distance for each point $Q_{i}$, ($i=1,2,3$)
from the basin boundary is demarcated by three lines (yellow, green, and red
lines, respectively). By our definition, the homogeneity index of each point
is then given by

$\displaystyle HI(Q_{1})$ $\displaystyle=$
$\displaystyle\left[\frac{3+2+2+1}{4}\right]^{-1}=2^{-1}=0.5$ $\displaystyle
HI(Q_{2})$ $\displaystyle=$
$\displaystyle\left[\frac{3+2+2+2}{4}\right]^{-1}=2.25^{-1}=0.44$
$\displaystyle HI(Q_{3})$ $\displaystyle=$
$\displaystyle\left[\frac{4+4+2}{3}\right]^{-1}=\left[\frac{10}{3}\right]^{-1}=0.3$

If all the squares boxes in a basin produce unimodal distributions, then
$HI(N_{i})=1\;\forall\;i$, and it is said to be a homogeneous basin. On the
other hand, if $HI(N_{i})<1$, the basin is inhomogeneous as it is found true
for the basin of our present study.

One can easily verify that the following two results are true for $HI$:

  * •

(i) $0\leq$ $HI$ $\leq 1$, and

  * •

(ii) along any fixed direction, $HI$ is a non-decreasing function of the
distance from the white basin.

It is obvious that $HI\geq 0$. To show that $HI$ of any point $Q$ cannot
exceed $1$, we note that

$\displaystyle
HI(Q)=\left[\frac{\Sigma(M(N_{i}))}{\mathbf{card}(N_{i})}\right]^{-1}$
$\displaystyle=$ $\displaystyle\frac{\mathbf{card}(N_{i})}{\Sigma(M(N_{i}))}$
$\displaystyle\leq$
$\displaystyle\frac{\mathbf{card}(N_{i})}{\mathbf{card}(N_{i})}~{}~{}$
$\displaystyle=$ $\displaystyle 1$

$[\because\mathbf{card}(N_{i})\leq\Sigma(M(N_{i})),~{}since~{}M(N_{i})\geq 1]$
To show the second result, we consider a line $L$, whose one end is on the
basin boundary. Now take any two points $G_{1},\,G_{2}$ on $L$ such that
$G_{2}$ is at a greater distance from the white basin than $G_{1}$. For the
sake of simplicity, we consider the reciprocal of the homogeneity index
$HI^{-1}$, which is essentially the arithmetic mean of the number of modes,
$\\{M(N_{i})\\}$, where $\\{N_{i}\\}$ are boxes through which $L$ passes to
reach the desired point. It becomes clear from Fig. 4(a), that for a
particular box $N_{i}$, $M(N_{i})$ remains constant or decreases as $N_{i}$
gets further away from the basin boundary. Thus the sequence $\\{M(N_{i})\\}$,
along any line, becomes a non-increasing sequence. Since we know that the
sequence of boxes leading to $G_{1}$ through $L$, is a subsequence of the
sequence leading to $G_{2}$, we infer
$\\{N_{i}\\}|_{G_{1}}\subseteq\\{N_{i}\\}|_{G_{2}}$ and hence
$\\{M(N_{i})\\}|_{G_{1}}\subseteq\\{M(N_{i})\\}|_{G_{2}}$. Now from the fact
that the arithmetic mean of a monotonically non-increasing sequence is also
non-increasing [33], we conclude

$HI(G_{1})^{-1}\geq HI(G_{2})^{-1}\implies HI(G_{2})\geq HI(G_{1}).$

Hence $HI$ is non-decreasing.

###  III.2 Anisotropy of distribution pattern of transients in the basin

Any pattern is called isotropic if the same properties are maintained at equal
distances in all directions from any fixed point of a space [33]. As observed
in Fig. 4(b), after traversing four square blocks along the yellow line unto
$Q1$, we arrive at a box of unimodal distribution of transient time. On the
other hand, traveling along the green line and covering an equal number of
square blocks, a bimodal distribution is encountered at $Q2$. This indicates a
directional dependence of the number of modes of distribution in the blue
basin, which appears as anisotropic. We attempt a quantification of this
structural property of the basin from the point of view of transient time.

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(a)

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(b)

Figure 5: Anisotropic pattern of distribution of transients in the basin. (a)
Two blue boxes are shown surrounded by immediate neighbors of yellow square
blocks called box neighborhoods. Red lines indicate the paths through which
the yellow regions could be reached from the blue boxes without passing
through any other square. (b) Map of the local isotropic index ($LII$). Dark
red (high locally isotropic) $0.8\leq LII\leq 1.0$, Red (moderate locally
isotropic) $0.6\leq LII<0.8$, Light red (poor locally isotropic) $LII<0.6$
(white color is not considered for calculation).

Before quantifying this isotropic property of a basin, we first define a box
neighborhood ($BN$). Let us consider a square block $N_{i}$, then the box
neighborhood, $BN(N_{i})$, of $N_{i}$ is defined as the square blocks $N_{j}$
$(j\neq i)$ such that $N_{j}$ can be accessed from $N_{i}$ without crossing
another box. In Fig. 5(a), the yellow squares around the two boxes at the
locations $[0.6,0.7]\times[0.2,0.3]$ and $[0.9,1.0]\times[0.9,1.0]$ represent
their box neighborhood.

Now we define a local isotropic index ($LII$) of a box $N_{i}$ as the
probability of finding square blocks in a box neighborhood of $N_{i}$ such
that they have the equal number of modes in their transient time distribution
as $N_{i}$. Thus, a $LII$ of the box $N_{i}$ is defined by

$\begin{split}LII(N_{i})=\frac{\mathbf{card}\\{N_{j}:j\neq i,N_{j}\in
BN(N_{i}),M(N_{j})=M(N_{i})\\}}{\mathbf{card}(BN(N_{i}))},\end{split}$ (4)

where $\mathbf{card}(X)$ is the cardinality of the set $X$.

For illustration purposes, we calculate the $LII$ of the two concerned boxes
in Fig. 5(a), namely $[0.6,0.7]\times[0.2,0.3]$ $(N_{1})$ and
$[0.9,1.0]\times[0.9,1.0]$ $(N_{2})$. We see from Fig. 4(a), $M(N_{1})=2$ and
$M(N_{2})=1$ and from Fig. 5(a), $BN(N_{1})=8$ and $BN(N_{2})=3$. Furthermore
from Fig. 4(a), the number of boxes with same number of modes as $N_{1}$ and
$N_{2}$ in their respective box neighborhoods are 8 and 1 respectively. Hence
by definition,

$\displaystyle LII(N_{1})$ $\displaystyle=$
$\displaystyle\left[{\frac{8}{8}}\right]=1$ $\displaystyle LII(N_{2})$
$\displaystyle=$ $\displaystyle\left[\frac{1}{3}\right]=0.33$

We now calculate the $LII$ of each $N_{i}$ and categorize the entire basin
into four groups listed in Table 3.

Table 3: Local isotropic indices Color | Description | Range
---|---|---
Dark red | highly locally isotropic | 0.8-1.0
Red | moderately locally isotropic | 0.6-0.8
Light red | poorly locally isotropic | 0.0-0.6
White | region not considered | -

0pt

(a)

0pt

(b)

0pt

(c)

Figure 6: Homogeneity index ($HI$) and local isotropic index ($LII$) against
initial resource level $R=0.9$. (a) Basin of attraction taking $R=0.9$. (b)
Variation in $HI$. Number of modes (indicated by a color bar) in each square
region when the basin is plotted on the plane $R=0.9$. (c) Distribution of
$LII$. Colors carry the same meaning as in Table 3.

###  III.3 Effect of resource enrichment

We check the variation in the distribution of the number of modes in the basin
with enrichment of resources with a different value of $R=0.9$ instead of
$R=0.5$ and plot $HI$ and $LII$ in Fig. 6. Quite understandably, the
inhomogeneity pattern changes with resource enrichment. Figure 6(a) shows that
the unimodal distribution of the transient time is a more dominant feature
instead of the dominant bimodal scenario in Fig. 4(a). However, on the left-
hand side of the white basin, a 6-mode distribution pattern is seen while for
the rest of the region, $M(N_{i})\mid_{R=0.5}\geq M(N_{i})\mid_{R=0.9}$. It
depicts that the same box in Fig. 4(a) produces a distribution with a higher
or equal number of modes than it does when the initial resource population
value is increased. This results in an increment of the average values of
$HI$.

On the contrary, we see no significant changes between Fig. 5(b) and Fig.
6(b). Since changing the resource plane from $R=0.5$ to $R=0.9$ only alters
the shape of the basins, $LII$ being a property independent of the location
and geometry of the basin boundary and only pertaining to the local box
neighborhood of the boxes ($N_{i}$), does not show any noticeable change.

##  IV Conclusion

In a classic resource-consumer-predator bistable food chain, the predator
shows extinction after an elapse of time called as transients when a state of
all coexisting species goes to a predator-free state after a transient time.
The distribution of the transients of predator extinction appears as
distinctively multimodal with a local variation in the number of modes that
appears as inhomogeneous and anisotropic relative to a basin boundary
separating two basins of the coexisting states. To be precise, the number of
modes as well as the mean transient time decreases with increase in the
distance from the basin boundary. We explained the reason behind the existence
of such distinct modes and the local variation in the number of modes and mean
transient time in the basin of a predator-free state from the dynamical nature
of the trajectories. We have quantified the characteristic structural
properties of the basin, in terms of the number of modes in distribution by
introducing two new metrics viz., homogeneity index and local isotropic index.
We witnessed an increasing homogeneity in the basin when the system is
enriched with resources.

The fate of the food chain is strongly influenced by the initial species
population. The food chain can either witness coexistence of all the species
or a state predator-free state after extinction. While considering the basin
of initial population densities that drive the top predator to extinction, we
observe existence of an intermediate state where all the three species
coexist, before the predator density finally becomes zero. The ecological
implications of such transient dynamics of species population and its
duration, are profound. The duration of existence of the intermediate state,
called transient time, is seen to have a clustering effect on the initial
population densities, where they are apparently divided into various
subpopulations, as evident from the multimodal distributions of transient time
[32]. Each subpopulation or group is seen to generate transient times equal to
or close to a particular principal value or mode. The entire set of initial
population densities, triggering predator extinction, can be divided into
various subsets or groups of suitable resolution. This division is
characterized here by two properties, inhomogeneity and anisotropy. From the
trend of inhomogeneity, we see that the initial population densities are
divided into fewer groups, as their difference with the critical initial
values, which decides two different fates of the ecosystem, increases. The
anisotropic factor reveals that the number of groups of species subpopulations
may change even if the difference between the critical and pertaining initial
population densities remains the same. Interestingly, when we enrich the
initial resource population in the food chain, the initial populations
densities become more homogeneous as reflected from the uniform distribution
of transients and are thus divided into fewer subgroups.

Extinction of species at any trophic level may cause the loss of energy flow
to the higher trophic levels, leading to a trophic cascade. Thus, species
persistence in a food structure is vital to prevent regime shift and ecosystem
functioning. Our analysis reveals that mean transient time is an increasing
multi-valued function of the number of modes in distribution. Broadly
speaking, the initial values generating multimodal distributions produce
trajectories with longer transient time than the ones producing unimodal
distributions. It tells us that there exists a wider time window for
protecting the predators from extinction using suitable restoration
mechanisms. Our initial study (see Appendix) indicates that the probability of
success of any restoration mechanisms is higher and the associated cost is
lower for the location of initial states leading to multimodal distribution
than in the unimodal one. A rigorous study is needed to ascertain which
restoration mechanisms is appropriate and cost-effective to save the
extinction of species with a higher probability. This is an open problem for
future study.

Acknowledgment: Research is supported by RUSA 2.0, Jadavpur University, Ref
No. R-11/725/19, dated 26.06.2019. A.M. is supported by the National Science
Centre, Poland, OPUS Programme Project No. 2018/29/B/ST8/00457.

Data availability: All data and numerical codes are available from the authors
free on request.

## Appendix

###  IV.1 Calculation of average spiraling time

0pt

(a)

0pt

(b)

0pt

(c)

0pt

(d)

Figure 7: Initial points are (0.5,0.65,0.85) for (a) and (0.5,0.95,0.85) for
(b). (a,b) Green curve corresponds to $T_{1}$, red curve to $T_{2}$ and blue
curve to $T_{3}=nT^{{}^{\prime}}$. (c,d) Poincaré return map of (a) and (b)
respectively, taking $R=0.87$ as the Poincaré plane. Parameters are chosen
from Table 1.

For numerical simulations, we have used the $4^{th}$ order Runge-Kutta
algorithm, taking temporal step size as 0.01. We elaborate here the
methodology for calculating the metrics required to quantify the different
segments ($T_{1},$ $T_{2}$ and $T^{{}^{\prime}}$) of the total transient time
($T$) for the Fig. 3(a,d) (see Section II.1). The calculation of the same for
other figures follows a similar manner.

The numerical solution of Eq. (1) is plotted in Fig. 7(a), such that the green
line represents the path of the trajectory from its initial point
(0.5,0.65,0.85) to the phase where it starts spiraling. On the other hand, the
blue line represents the spiraling phase of the trajectory. For calculating
the time required to reach the spiraling phase ($T_{1}$) and the subsequent
mean revolution time ($T^{{}^{\prime}}$), we consider the Poincaré return map
of Fig. 7(a), by taking the Poincaré plane as $R=0.87$. The plane is chosen so
that it cuts the curve almost at the point where the trajectory enters the
spiraling motion, demarcated by the change in the color of the trajectory from
green to blue. Thus, the time corresponding to the left-most dot in Fig. 7(c)
represents the end of the green curve of Fig. 7(a), that is $T_{1}$. The
considered dots correspond to the points where the curve cuts the Poincaré
plane after each spiral revolution and hence the time lapse between each dot
gives us the corresponding revolution time. For figuring out
$T^{{}^{\prime}}$, we note the times, say $F_{i},~{}i=1,2,3,....n$,
corresponding to each dot (red dots in Fig. 7(c)), where $n$ is the total
number of dots in Fig. 7(b). Observe that $n$ is $14$ here. Therefore, the
mean of the differences between the times corresponding to the adjacent dots
gives the mean revolution time, $T^{{}^{\prime}}$. Thus,

$T^{{}^{\prime}}=\frac{\sum_{i=1}^{n-1}(F_{i+1}-F_{i})}{n-1}.$

Fig. 7(b) is plotted in the same way as Fig. 7(a), but with initial points
(0.5,0.95,0.85), such that for this case $T_{2}\neq 0$. The calculation of
$T^{{}^{\prime}}$ from the Poincaré return map (Fig. 7(d)) is done in the same
manner as the previous case. For the determination of $T_{1}$ and $T_{2}$, we
simply note the time when the green curve of Fig. 7(b) hits the light yellow
plane $C=0.01$, thus obtaining $T_{1}$, and also note the same when the red
curve exists the yellow plane. The difference in the time of exiting and
entering the plane $C=0.01$ gives us $T_{2}$.

The magnitude of each metric concerning the calculation of transient time is
calculated by taking the mean of the results obtained after 50 simulations,
choosing random initial points from the pertaining colored box.

0pt

(a)

0pt

(b)

Figure 8: Effect of consumer induction: (a,b) Red and green curves indicate,
respectively, the predator behavior before and after the system is perturbed.
The perturbation is done with an addition of the consumer population density
of magnitude 0.01 in the consumer growth equation of the model.
(0.5,0.61,0.85) and (0.5,0.91,0.85) are the initial points of (a,b)
respectively. Parameter values correspond to Table 1.

###  IV.2 A possible restoration mechanism

It is earlier shown that predator extinction may be avoided in the food chain
by supplying additional predators to the system from outside [3]. Species
extinction can also be averted in this system by supplying additional
consumers to maintain predator’s growth (see Fig. 8). The required number of
consumers/predators for the survival of predator species varies with the
initial population densities, that is, the location of the basin. Fig. 8 shows
that a small number of additional consumers can prevent predator extinction
where the initial population is considered from the location with multiple
modes. However, the same number of consumers cannot protect the predator from
its death when the initial population is considered from a unimodal site of
the basin, which needs much higher additional consumers. This trend is
maintained throughout the basin.

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# A characterization of some highest weight modules for type A with
irreducible associated varieties

Zhanqiang Bai, Yixin Bao* and Xun Xie  School of Mathematical Sciences,
Soochow University, Suzhou 215006, P. R. China<EMAIL_ADDRESS>School of
Sciences, Harbin Institute of Technology, Shenzhen, 518055, P. R. China
<EMAIL_ADDRESS>School of Mathematics and Statistics, Beijing Institute
of Technology, Beijing 100081, China<EMAIL_ADDRESS>
###### Abstract.

For $\mathfrak{g}=\mathfrak{sl}(n,\mathbb{C})$, let $w$ be an element in the
Weyl group $W=S_{n}$. We use $L_{w}$ to denote a highest weight module of
$\mathfrak{sl}(n,\mathbb{C})$ with highest weight $-w\rho-\rho$. In this paper
we will give a characterization for those $w$ such that $L_{w}$ is a highest
weight Harish-Chandra module and the associated variety of $L_{w}$ will be
characterized by the information of $w$. We also count the number of those
highest weight Harish-Chandra modules $L_{w}$ in a given Harish-Chandra cell.
Finally, we give some other highest weight modules with irreducible associated
varieties from the smoothness of Schubert varieties.

Key Words: Highest weight module, Associated variety, Young tableau, Harish-
Chandara cell, Schubert variety.

###### 2010 Mathematics Subject Classification:

Primary 22E47; Secondary 17B08, 05A05

*Corresponding author

##  1\.  Introduction

Let $\mathfrak{g}$ be a finite-dimensional complex simple Lie algebra. In the
1960s, I. Gelfand and A. Kirillov [9] introduced an quantity to measure the
size of finitely generated $U(\mathfrak{g})$-module, which is now known as the
Gelfand-Kirillov dimension. For a finitely generated $U(\mathfrak{g})$-module
$M$, Vogan [19] constructed a variety $V(M)$ in $\mathfrak{g}^{\ast}$, which
is called the associated variety of $M$. Identifying $\mathfrak{g}^{\ast}$
with $\mathfrak{g}$ via the Killing form, the associated variety could also be
viewed as a variety in $\mathfrak{g}$. It is proved that the dimension of
$V(M)$ equals to the Gelfand-Kirillov dimension of $M$. Then the
characterization of associated varieties has risen as a significant topic of
infinite-dimensional representation theory of Lie algebras in recent years.

Fix a triangular decomposition
$\mathfrak{g}=\mathfrak{n}\oplus\mathfrak{h}\oplus\mathfrak{n}^{-}$ so that
$\mathfrak{h}$ is a Cartan subalgebra of $\mathfrak{g}$ and
$\mathfrak{b}=\mathfrak{h}+\mathfrak{n}$ is a Borel subalgebra. Let
$\mathbf{G}$ denote the algebraic adjoint group of $\mathfrak{g}$ and let
$\mathbf{B}$ denote the Borel subgroup of $\mathbf{G}$ corresponding to
$\mathfrak{b}$. For $\lambda\in\mathfrak{h}^{\ast}$, let $L(\lambda)$ be the
simple quotient of the Verma module $M(\lambda)$ with highetst weight
$\lambda$. Let $\rho$ be the half sum of positive roots. We denote $L_{w}$ the
simple highest weight module $L(-w\rho-\rho)$ with $w$ being an element in the
Weyl group $W$. Actually all simple highest weight modules of infinitesimal
character $\rho$ are of this form. To study the structure of the associated
varieties $V(L(\lambda))$, the following question naturally arises:

  * (Q1)

Can we find out all those weights $\lambda$ such that $V(L(\lambda))$ is
irreducible? In particular, can we find out all those $w$ such that $V(L_{w})$
is irreducible?

Actually, Joseph [11] proved that an associated variety $V(L_{w})$ is a union
of orbital varieties defined as follows. Let
$\mathcal{O}\subseteq\mathfrak{g}$ be a nilpotent $\mathbf{G}$-orbit. The
irreducible components of $\overline{\mathcal{O}}\cap\mathfrak{n}$ are called
orbital varieties of $\mathcal{O}$. They all take the form
$\mathcal{V}(w)=\overline{\mathbf{B}(\mathfrak{n}\cap w\mathfrak{n})}$ for
some $w\in W$. Then an associated variety of the form $V(L_{w})$ is
irreducible if and only if it contains only one orbital variety. Tanisaki [18]
showed that there exist reducible associated varieties of the form $V(L_{w})$
in type $B$ and $C$. For a long time, people conjectured that all associated
varieties of the form $V(L_{w})$ are irreducible in the case of type $A$ (see
[6] and [14]). However, Williamson [20] showed that there exist counter-
examples in 2014. So the structure of $V(L_{w})$ or $V(L(\lambda))$ is still
mysterious. In this paper, we will give some answers in the case of type $A$.
We classify the highest weight Harish-Chandra modules of the form $L_{w}$ and
describe their associate varieties explicitly. Furthermore, we also count the
number of such Harish-Chandra modules corresponding to a given associated
variety.

We refer [12] for the definition of Kazhdan-Lusztig right cell equivalence
relation and use $\stackrel{{\scriptstyle R}}{{\sim}}$ to denote the right
cell equivalence relation.

From now on we assume $\mathfrak{g}=\mathfrak{sl}(n,\mathbb{C})$. A weight
$\lambda=(\lambda_{1},\lambda_{2},...,\lambda_{n})$ is called integral if and
only if $\lambda_{i}\in\mathbb{Z}$ for $1\leq i\leq n$, and called
$(p,q)$-dominant if and only if $\lambda_{i}-\lambda_{j}\in\mathbb{Z}_{\geq
0}$ for $1\leq i<j\leq p$ and $p+1\leq i<j\leq p+q=n$. From Enright-Howe-
Wallach [8], we know the highest weight module $L(\lambda)$ is a Harish-
Chandra module if and only if $\lambda$ is $(p,q)$-dominant for some $p$ and
$q$. In particular, the weight $-w\rho-\rho$ is integral. Therefore, $L_{w}$
is a highest weight Harish-Chandra $SU(p,q)$-module if and only if
$-w\rho-\rho$ is $(p,q)$-dominant, if and only if $-w\rho$ is
$(p,q)$-dominant. The number of these modules is $|S_{n}/(S_{p}\times
S_{q})|=\frac{n!}{p!q!}$.

For a highest weight Harish-Chandra $SU(p,q)$-module, from Vogan [19] we know
that its associated variety is the closure of some $GL(p,\mathbb{C})\times
GL(q,\mathbb{C})$-orbit in

$\mathfrak{u}=\\{\left(\begin{array}[]{cc}0&c\\\ 0&0\\\
\end{array}\right)|c\in M(p,q,\mathbb{C})\\},$

where $M(p,q,\mathbb{C})$ denotes the set of complex $p$ by $q$ matrices. The
closures of $GL(p,\mathbb{C})\times GL(q,\mathbb{C})$-orbits in $\mathfrak{u}$
form a linear chain of varieties:

(1.1)
$\\{0\\}={\bar{\mathcal{O}}}_{0}\subset\bar{\mathcal{O}}_{1}\subset...\subset\bar{\mathcal{O}}_{r-1}\subset\bar{\mathcal{O}}_{r}=\mathfrak{u},$

where $r=\min\\{p,q\\}$. From NOTYK [15], we know

$\mathcal{O}_{j}=\\{\left(\begin{array}[]{cc}0&c\\\ 0&0\\\
\end{array}\right)|c\in M(p,q,\mathbb{C}),\text{rank}(c)=j\\}.$

And we have

$\overline{\mathcal{O}_{m}}=\coprod\limits_{j\leq
m}\mathcal{O}_{j}=\\{\left(\begin{array}[]{cc}0&c\\\ 0&0\\\
\end{array}\right)|c\in M(p,q,\mathbb{C}),\text{rank}(c)\leq j\\}.$

Sometime we will write it as $\overline{\mathcal{O}_{m}}(p,q)$ to emphasize
that it comes from $SU(p,q)$. From Bai-Xie [2] or NOTYK [15], we know
$\dim{\mathcal{O}_{j}}=j(n-j)$.

Throughout this paper, we identify an element $w$ in the Weyl group $S_{n}$
with the $n$-tuple $(w(1),w(2),\cdots,w(n))$, where $w(i)$ is the image of $i$
under $w$. Suppose $p\geq q$ and $p+q=n$. An element $w\in S_{n}$ is called
$(p,q)$-decreasing if $w=(x_{p},...,x_{1},y_{q},...,y_{1})$ with
$x_{p}>...>x_{1}$, $y_{q}>...>y_{1}$ and $x_{1}<y_{1}$,…,$x_{q}<y_{q}$.

In this paper, we give a characterization of those $w$ such that $L_{w}$ is a
Harish-Chandra module. We will prove the following theorem.

######  Theorem 1.1.

If $L_{w}$ is a Harish-Chandra $SU(p,q)$-module, then $w$ is right cell
equivalent to a $(n-m,m)$-decreasing element $w_{p,q,m}$, where

$w_{p,q,m}=(n,n-1,...,p+m+1,p,p-1,...,1,p+m,p+m-1,...,p+1)$

and the integer $0\leq m\leq\min\\{p,q\\}$ is uniquely determined by $w$. We
also have

$V(L_{w})=\mathcal{V}(w)=\overline{\mathcal{O}_{m}}(p,q).$

Conversely, if $w\stackrel{{\scriptstyle R}}{{\sim}}w_{p,q,m}$ for some
integer $m$ between $0$ and $\min\\{p,q\\}$, then $L_{w}$ is a highest weight
Harish-Chandra $SU(p,q)$-module.

Notice that $w_{p,q,m}$ is reduced to $(p,p-1,\cdots,1,n,n-1,\cdots,p+1)$ in
Theorem 1.1 if $p+m+1>n$ (equivalently $p\geq q=m$). In this case, we have
$w_{p,q,m}=w_{p,q,m}^{-1}$.

Based on Theorem 1.1 and the famous hook formula (see James [10]), we obtain
the main result in Section 3 as follows:

######  Theorem 1.2.

The number of Harish-Chandra modules $L_{w}$ such that
$V(L_{w})=\overline{\mathcal{O}_{m}}(p,q)$ is

$\left\\{\begin{array}[]{ll}\frac{n(n-1)...(n-m+2)(n-2m+1)}{m!},&\hbox{if
$2\leq m\leq\min\\{p,q\\}$;}\\\ n-1,&\hbox{if $m=1$;}\\\ 1,&\hbox{if
$m=0$.}\end{array}\right.$

In the case of type $C$, Barchini-Zierau [5] have computed the size of each
Harish-Chandra cell.

From this theorem we can get the following interesting corollary.

######  Corollary 1.1.

For any positive integers with $p\geq 2$, $q\geq 2$ and $p+q=n$, we have

$n+\sum\limits_{2\leq m\leq
min(p,q)}\frac{n(n-1)...(n-m+2)(n-2m+1)}{m!}=\frac{n!}{p!q!}.$

In the 1980s, Borho-Brylinski [6] found that for smooth Schubert varieties
$X_{w}=BwB/B$, we will have $V(L_{w})=\mathcal{V}(w)$. Later on, Lakshmibai-
Sandhya found a criterion [13] to determine the smoothness of $X_{w}$. By
using these results, we can find some other highest weight modules $L_{w}$
with irreducible assiciated varieties.

##  2\.  Proof of the theorem 1.1

We will prove Theorem 1.1 in this section. Sagan [16] established a bijection
between the symmetric group $S_{n}$ and the set of pairs of standard Young
tableaux of size $n$ through the famous Robinson-Schensted algorithm. We use
$P(\sigma)$ (resp. $Q(\sigma)$) to denote the corresponding insertion (resp.
recording) Young tableau for any $\sigma\in S_{n}$. Note that
$Q(\sigma)=P(\sigma^{-1})$. Then we may define the left, right and double cell
equivalence relations on the Weyl group $S_{n}$ through this bijection (see
Ariki [1] and Steinberg [17]). For example, $\sigma\stackrel{{\scriptstyle
R}}{{\sim}}w$ if and only if $P(\sigma)=P(w)$.

To prove our theorem, we recall some properties about orbital varieties.

######  Proposition 2.1 (Steinberg [17]).

For any $w,y\in W=S_{n}$, we have $\mathcal{V}(w)=\mathcal{V}(y)$ if and only
if $w\stackrel{{\scriptstyle R}}{{\sim}}y$.

######  Proposition 2.2 (Joseph [11]).

For $\mathfrak{g}=\mathfrak{sl}(n,\mathbb{C})$ and $W=S_{n}$, we have
$\mathcal{V}(w)\subseteq V(L_{w})$ and $\dim\mathcal{V}(w)=\dim
V(L_{w})=GKdim(L_{w})$.

Proof of Theorem 1.1. We first prove that if $L_{w}$ is a highest weight
Harish-Chandra $SU(p,q)$-module, then $w$ is right cell equivalent to

$w_{p,q,m}=(n,n-1,...,p+m+1,p,p-1,...,1,p+m,p+m-1,...,p+1)$

for some $0\leq m\leq\min\\{p,q\\}$. The uniqueness of $m$ is obvious since
$P(w_{p,q,m_{1}})=P(w_{p,q,m_{2}})$ if and only if $m_{1}=m_{2}$. Then we only
need to prove the existence of the integer $m$. Denote $W_{p,q}$ the subset of
$S_{n}$ consisting of elements $w$ such that $-w\rho$ are $(p,q)$-dominant.
Obviously, a highest weight module $L_{w}$ is a highest weight Harish-Chandra
$SU(p,q)$-module if and only if $w\in W_{p,q}$. We prove the existence of $m$
in two steps. In the first step, we construct a $(n-m,m)$-decreasing element
$z$ in $S_{n}$ which is right cell equivalent to a given element $w\in
W_{p,q}$. In the second step, we prove that $z=w_{p,q,m}$.

For the first step, the process of constructing $z$ can be described as
follows. Write $-w\rho=(t_{1},...,t_{p},t_{p+1},...,t_{n})$, which is
$(p,q)$-dominant and $\lambda=-w\rho-\rho$. Let $m$ be the maximum nonnegative
integer for which there exists a sequence of indices

$1\leq i_{1}<i_{2}<...<i_{m}\leq p<p+1\leq j_{m}<...<j_{1}\leq n$

such that

$t_{i_{1}}-t_{j_{1}}\leq 0,t_{i_{2}}-t_{j_{2}}\leq
0,...,t_{i_{m}}-t_{j_{m}}\leq 0.$

Then from Bai-Xie [2], we can get a Young tableau $P(\lambda)$ with two
columns or one column (for $w=-Id$). We only consider the nontrivial case.
Then from Bai-Xie [2], we know $P(\lambda)$ and $P(w)$ have the same shape
(actually $P(\lambda)=P(w^{-1})$). From the construction of $P(\lambda)$, we
know the number of entries in the second column of $P(\lambda)$ equals to $m$.
Suppose the entries in the first column of $P(w)$ are $(s_{1},...,s_{n-m})$
and the entries in the second column of $P(w)$ are $(a_{1},...,a_{m})$ from
top to bottom. Then we get $w\stackrel{{\scriptstyle
R}}{{\sim}}(s_{n-m},...,s_{1},a_{m},...,a_{1}):=z$, while $z$ is
$(n-m,m)$-decreasing. Before we go to the second step, from Bai-Xie [2] we
have

$V(L_{w})=\overline{\mathcal{O}}_{m}(p,q)=\mathcal{V}(w)=\mathcal{V}(z).$

In the second step, we take a special element

$\sigma_{p,q,m}=(n-q,...,n-q-m+1,n,...,n-q+1,p-m,...,1).$

Note that if we write $-\rho=(b_{1},...b_{n})$, then

$-\sigma_{p,q,m}\rho=(b_{n},b_{n-1},...,b_{m+q+1},b_{m},b_{m-1},...b_{1},b_{m+q},...,b_{2m},...b_{m+1})$

is $(p,q)$-dominant. So $\sigma_{p,q,m}\in W_{p,q}$. Applying the process in
the first step to $\sigma_{p,q,m}$, it holds that $\sigma_{p,q,m}$ is right
cell equivalent to $w_{p,q,m}$. From Bai-Xie [2], we know that

$V(L_{\sigma_{p,q,m}})=\overline{\mathcal{O}}_{m}(p,q)=\mathcal{V}(\sigma_{p,q,m})=\mathcal{V}(w_{p,q,m}).$

Then $\mathcal{V}(z)=\mathcal{V}(w_{p,q,m})$. By Proposition 2.1, we have
$z\stackrel{{\scriptstyle R}}{{\sim}}w_{p,q,m}$ and $z=w_{p,q,m}$.

Conversely, suppose $w\stackrel{{\scriptstyle R}}{{\sim}}w_{p,q,m}$ for some
$0\leq m\leq\min\\{p,q\\}$. We need prove that $L_{w}$ is a highest weight
Harish-Chandra $SU(p,q)$-module. We recall the concept of Harish-Chandra
cells, which is defined by Barbasch-Vogan [3]. Let $X,Y$ be two irreducible
(Harish-Chandra) modules with the same infinitesimal character. We write $X>Y$
if there exists a finite-dimensional $\mathfrak{g}$-module $F$ such that $Y$
appears a sub-quotient of $X\otimes F$. We write $X\sim Y$ if both $X>Y$ and
$Y>X$. The equivalence classes for the relation $\sim$ are called cells of
(Harish-Chandra) modules or (Harish-Chandra) cells. From Barbasch-Vogan [4],
we know two highest weight modules $L_{w}$ and $L_{y}$ belong to the same
(Harish-Chandra) cell with infinitesimal character $\rho$ if and only if
$w\stackrel{{\scriptstyle R}}{{\sim}}y$. It follows from this definition that
if a cell contains one highest weight Harish-Chandra module, then all
representations in this cell are highest weight Harish-Chandra modules.

So we only need to show that $L_{w_{p,q,m}}$ is a highest weight Harish-
Chandra $SU(p,q)$-module. Equivalently, we need to show that $-w_{p,q,m}\rho$
is $(p,q)$-dominant. By a direct calculation, we have $-w_{p,q,m}\rho=$

$\frac{1}{2}(n-2m-1,n-2m-3,\cdots,n-2m-2p+1,n-1,n-3,\cdots,n-2m+1,n-2m-2p-1,\cdots,1-n).$

It is easy to check that $-w_{p,q,m}\rho$ is $(p,q)$-dominant. Then we finish
the proof of Theorem 1.1.

######  Corollary 2.1.

The highst weight module $L_{w}$ is a Harish-Chandra $SU(p,q)$-module and
$V(L_{w})=\mathcal{V}(w)=\overline{\mathcal{O}}_{m}(p,q)$ if and only if

$w\stackrel{{\scriptstyle
R}}{{\sim}}w_{p,q,m}=(n,...,p+m+1,p,...,1,p+m,...,p+1).$

Note that when $m=0$, we will have $-w\rho-\rho=0$ and $w=-Id$.

From this corollary, we can see that there is only one Harish-Chandra cell
with associated variety $\overline{\mathcal{O}}_{m}(p,q)$.

##  3\.  The size of highest weight Harish-Chandra cells

Recall the concept of Harish-Chandra cells in Section 2. In the case of type
$C$, Barchini-Zierau [5] have computed the size of each Harish-Chandra cell
consisting of highest weight Harish-Chandra modules with a given associated
variety. In the case of type $A$, we have the following theorem.

######  Theorem 3.1.

We use $\mathcal{C}_{m}$ to denote a Harish-Chandra cell consisting of highest
weight Harish-Chandra $SU(p,q)$-modules $L_{w}$ with associated variety
$\overline{\mathcal{O}}_{m}(p,q)$, then for $2\leq m\leq\min\\{p,q\\}$

$\displaystyle\\#\mathcal{C}_{m}$ $\displaystyle=\text{the number of standard
Young tableaux of shape~{}}P(w)$
$\displaystyle=\frac{n(n-1)...(n-m+2)(n-2m+1)}{m!}.$

And $\\#\mathcal{C}_{0}=1$, $\\#\mathcal{C}_{1}=n-1$.

###### Proof.

From the proof of our theorem 1.1, we know the number of modules in the
Harish-Chandra cell $\mathcal{C}_{m}$ equals the number of elements in the
right cell $\mathscr{C}_{m}$ which contains $w_{p,q,m}$. On the other hand,
$\\#\mathscr{C}_{m}$ equals the number of standard Young tableaux of shape
$P(w_{p,q,m})$. We know the numbers of entries in the two columns of
$P(w_{p,q,m})$ are $c_{1}(P(w_{p,q,m}))=n-m$ and $c_{2}(P(w_{p,q,m}))=m$. From
James [10], the corresponding hooks of $P(w_{p,q,m})$ is a Young tableau
$P_{m}$ consisting of two columns. The entries in the first column of $P_{m}$
are $(n-(m-1),n-m,...,n-2m+2,n-2m,...,1)$ and the entries in the second column
of $P_{m}$ are $(m,m-1,...,1)$ from top to bottom. From the famous hook
formula (see James [10]), we have

$\displaystyle\\#\mathscr{C}_{m}$
$\displaystyle=\frac{n!}{(n-(m-1))\times...\times(n-2m+2)\times(n-2m)\times...\times
1\times m\times...\times 1}$
$\displaystyle=\frac{n(n-1)...(n-m+2)(n-2m+1)}{m!}.$

When $m=1$, we will have $c_{1}(P(w_{p,q,m}))=n-1$ and
$c_{2}(P(w_{p,q,m}))=1$, so

$\\#\mathcal{C}_{1}=\frac{n!}{n\times(n-2)\times...\times 1\times 1}=n-1.$

When $m=0$, we will have $c_{1}(P(w_{p,q,m}))=n$ and $c_{2}(P(w_{p,q,m}))=0$,
so $\\#\mathcal{C}_{0}=1$.

∎

######  Remark 3.1.

From Barchini-Zierau [5], we know only highest weight Harish-Chandra
$SU(p,q)$-modules can have associated variety
$\overline{\mathcal{O}}_{m}(p,q)$. So there is only one Harish-Chandra cell
$\mathcal{C}_{m}$ of irreducible Harish-Chandra $SU(p,q)$-modules with
associated variety $\overline{\mathcal{O}}_{m}(p,q)$.

##  4\. Some other highest weight modules with irreducible associated
varieties

In this section, we will find some other highest weight modules which are not
Harish-Chandra modules and they have irreducible associated varieties. First
we need some propositions about Schubert varieties.

######  Proposition 4.1 (Borho-Brylinski [6]).

For $\mathfrak{g}=\mathfrak{sl}(n,\mathbb{C})$ and $W=S_{n}$, if the Schubert
variety $X_{w}=\overline{BwB/B}$ is smooth, we will have
$V(L_{w})=\mathcal{V}(w)$.

An element $w\in S_{n}$ contains a pattern $\mu\in S_{k}$ if there is a
subword with $k$ letters in $w$ with the same relative order of the letters as
in $\mu$ . We say $w$ avoids the pattern $\mu$ if $w$ does not contain this
pattern. We have the following criterion for smoothness of Schubert varieties.

######  Proposition 4.2 (Lakshmibai-Sandhya ).

For $\mathfrak{g}=\mathfrak{sl}(n,\mathbb{C})$ and $W=S_{n}$, the Schubert
variety $X_{w}=\overline{BwB/B}$ is smooth if and only if $w$ avoids the two
patterns $3412$ and $4231$.

So if $w$ avoids the two patterns $3412$ and $4231$, the associated variety of
$L_{w}$ will be $\mathcal{V}(w)$.

######  Proposition 4.3 (Carrell [7]).

For $\mathfrak{g}=\mathfrak{sl}(n,\mathbb{C})$ and $W=S_{n}$, the Schubert
variety $X_{w}=\overline{BwB/B}$ is smooth if and only if $X_{w^{-1}}$ is
smooth.

Borho-Brylinski [6] also found that $V(L_{w})$ is irreducible if and only if
$V(L_{w^{-1}})$ is irreducible.

From the the above propositions we will have the following corollary.

######  Corollary 4.1.

Suppose $L_{w}$ is a highest weight Harish-Chandra $SU(p,q)$-module and
$V(L_{w})=\mathcal{V}(w)=\overline{\mathcal{O}}_{m}(p,q)$. If
$\sigma\stackrel{{\scriptstyle R}}{{\sim}}(n-m,...,1,n,...,n-m+1)$, we will
have $V(L_{\sigma})=V(L_{w_{p,q,m}^{-1}})=\mathcal{V}(w_{p,q,m}^{-1})$.

###### Proof.

When $L_{w}$ is a highest weight Harish-Chandra $SU(p,q)$-module, we know
$V(L_{w})=\mathcal{V}(w)=\overline{\mathcal{O}}_{m}(p,q)$ if and only if
$w\stackrel{{\scriptstyle
R}}{{\sim}}w_{p,q,m}=(n,...,p+m+1,p,...,1,p+m,...,p+1)$. Thus we have

$\displaystyle w_{p,q,m}^{-1}$
$\displaystyle=(n-m,...,n-(p+m)+1,n,...,n-m+1,n-(p+m),...,1)$
$\displaystyle\stackrel{{\scriptstyle R}}{{\sim}}(n-m,...,1,n,...,n-m+1).$

Now $\sigma\stackrel{{\scriptstyle
R}}{{\sim}}(n-m,...,1,n,...,n-m+1)\stackrel{{\scriptstyle
R}}{{\sim}}w_{p,q,m}^{-1}$. So from our theorem and proposition 4.1 and 4.3,
we have $V(L_{\sigma})=V(L_{w_{p,q,m}^{-1}})=\mathcal{V}(w_{p,q,m}^{-1})$. ∎

Suppose $w=(s_{n-m},...,s_{1},a_{m},...,a_{1})$ is an element of
$(n-m,m)$-decreasing in $W=S_{n}$. We take out the largest $a_{k}$ satisfying
$s_{n-m}>a_{k}$ and the smallest $s_{l}$ satisfying $s_{l}>a_{1}$. If
$s_{l}>a_{k}$, we will have $s_{n-m}>s_{l}>a_{k}>a_{1}$ and such a $w$ will be
called _a good full element of size $(n-m,m)$_. Note that $a_{k}$ and $s_{l}$
may not exist. We still call such $w$ a good full element of size $(n-m,m)$.
Note that the good full elements of size $(n-m,m)$ and the $w_{p,q,m}$ in our
theorem have the same double cell. So we have the following corollary.

######  Corollary 4.2.

Suppose $\mathfrak{g}=\mathfrak{sl}(n,\mathbb{C})$ and $W=S_{n}$. If
$w\stackrel{{\scriptstyle
R}}{{\sim}}w_{0}=(s_{n-m},...,s_{1},a_{m},...,a_{1})$, where $w_{0}$ is some
good full element of size $(n-m,m)$, we will have $V(L_{w})=\mathcal{V}(w)$
and $V(L_{w^{-1}})=\mathcal{V}(w^{-1})$.

###### Proof.

Note that the given $w_{0}=(s_{n-m},...,s_{1},a_{m},...,a_{1})$ is a good full
element of size $(n-m,m)$. From its definition we know it avoids the two
patterns $3412$ and $4231$. So $V(L_{w})=\mathcal{V}(w)$ by the proposition
4.2 and 4.1 and $V(L_{w^{-1}})=\mathcal{V}(w^{-1})$ by the proposition 4.3.

∎

######  Remark 4.1.

There may exist some $w\in W=S_{n}$ such that $X_{w}$ is not smmooth but
$V(L_{w})=\mathcal{V}(w)$. For example, when $n\leq 6$, we have
$V(L_{w})=\mathcal{V}(w)$ for all $w\in W=S_{6}$. But in $S_{6}$ there are
many permutations containing the pattern $3412$ or $4231$.

### Acknowledgments

The first author is supported by the National Science Foundation of China
(Grant No.11601394), the second author is supported by the National Science
Foundation of China (Grant No.11801117) and the Natural Science Foundation of
Guangdong Province, China (Grant No.2018A030313268), the third author is
supported by the National Science Foundation of China (Grant No.11801031 and
No.11601116).

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