Abstract: - Invasive species management has developed into a highly specialized field utilizing a systems
approach. It requires knowledge of their life history, growth requirements, and population dynamics that integrate
their biology and control. The foundation of strategic planning for the management of invasive species is laid
by demographic studies, which record the birth, growth, reproduction, and death of individuals within a
population. The present study makes use of the Discrete Leslie Matrix Model to analyze the growth in the age-
structured population of Euphorbia hirta, an invasive species in agrosystems, identify critical stages in the
species' life cycle, and project the structure and size of future population.
Key-Words: Invasive species, Leslie Matrix, critical stage, management, future population
Received: November 28, 2023. Revised: June 19, 2024. Accepted: July 12, 2024. Published: August 12, 2024.
1 Introduction
Invasive alien plant species are those that have been
intentionally or unintentionally brought outside of
their native habitat. They have an impact on human
safety, habitats, biodiversity, ecology, and spread
out of control [1; 2]. A biological invasion has been
identified as one of the main causes of economic
and environmental disruption, and biodiversity
loss. Compared to native plants, invasive alien
plants have advantages such as faster growth, greater
photosynthetic rates, higher reproductive output,
more biomass, lower carbon-to-nutrient ratios in
tissue, stronger nutrient absorption capacities, and
higher plasticity levels [3].
For efficient management and population regulation
of invasive species, one must have a thorough
understanding of the ecology, morphology,
reproductive biology, physiology, and biochemistry
of such species as a wide variety of factors regulate
the density, growth, and competitive ability of these
plants. According to Funk [4], invasive species have
a greater strategic advantage for nutrient use over
native plants. They are also more common along
roadside and places with anthropogenic
disturbances [5]. Plant invasions are detrimental to
ecology and global biodiversity changing the
landscapes. Many invasive weed species have
invaded terrestrial crops worldwide [6] and
drastically lowered agricultural output. Species
invasions are a major component in global change
resulting in habitat degradation, altering the
biological diversity and environmental mechanism,
causing extinction of local flora and fauna,
modifying ecosystem functioning and services, and
promoting subsequent invasions that exacerbate the
damage. [7;8;9; 10;11]. Climate change and
biological invasions represent two of the largest
threats to biodiversity in the Anthropocene [12].
Early detection of invasive plants, can help in weed
management by efficient eradication and billions of
dollars for ongoing control to stem biodiversity loss
[13].
Adaptation of weed management strategies reduce
the incidence of invasive species, decrease
their undesirable effects, and optimize land use thro
ugh the combination of preventive and control
practices [14].
Invasive alien plant species may have a major effect
on global agriculture, which continues to have an
impact on food security worldwide [15]. Globally,
invasive weed species need to be controlled via
mechanical, physical, biological, and chemical
methods. Many invasive plant species do not have
any biological control agents [16]. Climate change
[17], deforestation, ecological degradation, and
anthropogenic disturbances all worsen agricultural
production worldwide.
1.1 Demographic models
Mathematical models significantly contribute to the
prediction the spread of invasive species and
directing the optimal allocation of resources for their
prevention, control, or eradication [18;19].
Models are indispensable tools for managing
Leslie Matrix Model For Euphorbia Hirta L Population
ASHA GUPTA
Department of Life Sciences
Centre of Advanced Study in Life Sciences
Manipur University, Canchipur-795003,
INDIA
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DOI: 10.37394/232029.2024.3.12
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E-ISSN: 2945-0454
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Volume 3, 2024
invasive species. According to Baker and Bode [20],
scientists can utilize them to calculate vital rates
like the rate of spread, model the possible impacts of
invasive species, and investigate the consequences
of different strategies for controlling or eliminating
them.
Introduced species may experience climatic niche
shifts when moving to new continents because of
changes to their fundamental or realized niche [21].
Whereas niche shifts are evaluated in ‘climatic
space’—usually with ordination techniques—the
general goal of species distribution models (SDMs)
is to project climatic niche models into geographic
space using Matrix or other predictive models.
Demographic models are useful for understanding
population processes and stages in the life cycle of
the species that could be most effectively targeted
with management. The study carried out by Bogdan
et al [22] in demographic data of an Israeli
Carpobrotus population gives an Integral Projection
Model and through the analysis of asymptotic
growth rates and population sensitivities and
elasticities demonstrated the population as stable,
and reducing the survival of the largest individuals
reduced the overall population growth.
Chung et al [23], developed an integrated spatial
model to manage common ragweed (Ambrosia
artemisiifolia var. elatior (L.,) Decs) using various
models, including species distribution BIOMOD2,
landchange LCM, dispersal MigClim and
optimization model prioritize and proposed a new
'removal effect index' for evaluation in time series.
Guetling et al [24], developed habitat susceptibility
models (HSM) created within geographic
information systems (GIS) to combine spatial
environmental data at known infestations and
predict areas likely to be invaded based on similar
ground conditions [25] for meadow hawkweed and
orange hawkweed. With known locations and
environmental data, the predictive models were used
to estimate habitat susceptibility for invaders by
determining the indicator species.
Species distribution models (SDMs) are often used
to produce risk maps to guide conservation
management and decision-making with regard to
invasive alien species (IAS). Davis et al [26]
developed WiSDM, a semi-automated workflow to
democratize the creation of open, reproducible,
transparent, invasive alien species risk maps.
Worldwide, it is generally acknowledged that
invasive species pose a serious threat to native
biodiversity, ecosystem function, and economic
interests at a global scale [27]. Structured population
models (like matrix population models [28], integral
projection models [29] of invasive plants provide a
tool for producing comprehensive fitness estimates
and identify sensitive vital rates (e.g. survival,
growth) to target with management [28;30;31].
1.2 Population Dynamics of Invasive species
Invasive species management is concerned with
maximizing mortality and lowering the reproduction
and minimizing the loss of resources resulting from
invasive species competition. In this context the role
of natural and man-managed factors that regulate the
size of invasive species population becomes of
paramount importance. A classification of
individuals by age in such a population provides
reasonably accurate prediction of their demographic
potential. Population dynamics of species helps in
analyzing patterns related to growth, reproduction
and mortality theoretically in mathematical terms.
Demography studies help to learn empirically how
the population grows in nature. They deal it by
keeping the track of the birth, growth, reproduction
and death of individuals in a population. It can then
form the basis of strategic planning for invasive
species control [32;33].
1.3 Matrix population models
Matrix population models are categorized into two
types of models that are in vogue viz. Age
structured models described by Leslie
[34] and stage structured models described by
Lefkovitch[35].
Sensitivity and elasticity assessments are also
performed using matrix population models [36;37].
When an individual's attribute other than age is a
stronger predictor of survival and reproduction,
then according to Caswell [28] and Cochran and
Ellner [38] the models that are preferred are stage
structured matrix models.
In numerous research [39; 40;41; 42], vital rates
in stage-structured population models can be
inferred from age-structured vital rates, where age
classes are grouped together to form a stage.
There has been an upsurge in the use of stage-
structured population models, and a life table
analysis is often used to estimate the vital rates in
these models [43]. The relationship between the
number of stages and different statistics derived
from stage-structured population matrices is also
covered by Salguero-Gomez & Plotkin [44].
Lebreton [45] suggests using models that are stage-
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structured, meaning that Stages are embedded
within the age classes. The dynamics of stage-
structured populations have been described by
means of both stochastic and deterministic models.
In her contribution. Pasquali [46] focuses on stage-
structured demographic models, in which growth of
an individual is described by its physiological age,
which is governed by a stochastic differential
equation.
1.4 INTEGRATING POPULATION
DYNAMICS AND POPULATION
STRUCTURE
Matrix population models that integrate population
dynamics and population structure are a power tool
for investigating population dynamics [47].
It has also been long known that age specific effects
have a profound influence on overall population
dynamics [48]. Matrix projection model based on
age specific characteristics of individuals
[49;50;34], have become a means of characterizing
populations and predicting their future behaviour
[51]. In some areas of resource biology, the
sampling programmes are mobilized in order to
obtain data to build life tables from which
population dynamics models may be developed. The
primary objectives of population dynamics
modelling are twofold; to give insight into the
biological mechanisms operating in the system
being modelled and to produce a model of
community interactions which predicts changes in
abundances (numbers of community species). In
recent years, there has been a remarkable expansion
in the application of matrix models. Quantitative
demographic analysis should be used more
frequently to advise management, according to
population ecologists [52;53;54].
The Discrete Leslie Matrix Model [55;34;56] has
been used to analyze the growth in age structured
populations. Matrix population models that integrate
population dynamics and population structure are a
power tool for investigating population dynamics
[47]. It has also been long known that age specific
effects have a profound influence on overall
population dynamics [57].
In this study, the Discrete Leslie Matrix Model
[55;34;56] has been used to analyse the growth in
age structured population of Euphorbia hirta Linn,
an invasive species in agrosystems with the
following objectives:
i) to study the temporal dynamics of target
weed population
ii) to investigate the transient dynamics and
asymptotic characteristics of the population
iii) to project the structure and size of future
populations and
(iv) to suggest the critical stages in the life cycle
of the weed for management programme
2 Material and Methods
2.1 The Species
Euphorbia hirta Linn.selected for the present study,
commonly known as Pill-bearing spurge, is a
member of the Euphorbiaceae family. It is a little
annual herb that is propogated by seeds. The species
is currently found throughout tropical and
subtropical regions, having originated in Tropical
America. It is commonly observed inhabiting paths,
roadside vegetation, grasslands, banks of
watercourses, and open waste areas [58]. Due to
increased trade, tourism, industry growth,
transportation, technology advancements, and rising
rates of urbanization, the invasion of species has
expanded significantly [59]. According to Pauchard
et al. [60], the invasion of various regions by alien
species increase pressure on the natural
environment.
Euphorbia hirta is an important agrestal weed with
special affinity for paddy tracts, irrigated and garden
crops. It has been documented that Euphoria hirta
infestations occur in rice, mung-sesame system,
chilli, maize, and mustard (B. juncea) crops [61;62;
63]. Aqueous extract of E. hirta at high
concentration was found to impair the growth of
maize and wheat seedlings, delayed germination,
reduced chlorophyll and wheat protein content [63].
It exerts allelopathic effect on crops like potato,
sugarcane, maize and sorghum by competing with
native plants and discouraging grazing near it, the
plant reduces forage production and interferes with
pasture/rangeland and livestock [64]. This directly
affects the land's suitability for livestock grazing.
Weeds on rangelands have an adverse effect on the
livestock industry by reducing forage supply and its
quality, obstructing grazing, poisoning animals,
raising the expense of managing and producing
livestock, and decreasing land value [65;66]. In
pastures and rangelands, E. hirta easily
outcompetes desirable vegetation [67].After
establishing itself in pasture and rangeland habitats,
it tends to displace all other vegetation
[68;69].Found widely in moist and dry
environments, E. hirta creates through allelopathy,
essentially a single species stand by releasing the
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flavonoid compound Kaempherol glucuronide, the
plant is poisonous to animals and presents a
significant risk to the productivity of livestock on
open rangelands [70]. E. hirta is selected for the
present study.
2.2 Study Area:
The study was conducted around Imphal (24o75' N
latitude and 93o85' E Longitude at an elevation of
782 m MSL) at Imphal valley, Manipur, North East
India. The average maximum temperature ranges
from 25.1oC-31.1oC (May - June) while the
minimum temperature ranges from 11.8oC to
19.4oC. (December-January), the average annual
rainfall being 1470 mm per annum. Humidity is
highly variable during different seasons ranging
from 45 to 100%.
2.3 Demographic Analysis:
Over two-year period, field study was conducted on
pure natural population of forty randomly chosen
0.25m-2 plots demarcated for studying the
demographic parameters. The number of individuals
in 4 different functional stages called age groups
were recognized in the field conditions and marked.
These 4-age group comprised of seedling, juvenile,
flowering adults and fruiting adults. Record of the
individual plants in the above said area for each
stage was made. New individuals becoming
established at the time of each census were identified
as per age group. The number of individuals in each
age group were censused at monthly intervals and
the observations were continued for more than 2
years (December 2020-December 2022). To assess
the impact of nutrient supply on seedling emergence,
soil samples were taken at 2-6 cm depth from ten
adjacent plots and analyzed for Soil Moisture, pH,
Organic Carbon, Total Nitrogen, Available
Phosphorus and Exchangeable Potassium as per the
standard methods given in Misra [71].
2.4 The Model
The equation,
Av = λv
indicates that A = square matrix
ν = column vector and
λ = scalar.
For every Eigen value λ, there is an associated Eigen
vector ν. The dominant Eigen value gives the rate at
which the population size increases.
The Leslie matrix model take the general form of
N(t+1) = A.N (t).
Where, N(t+1) and N(t) are vectors representing age
or stage class distribution of individuals at time t+1
and t, A is the matrix defining the age or stage
specific survival and fecundity values. The rate of
emigration and immigration of propagules were
assumed to be equal. The model was thus simplified.
The asymptotic population growth rate λ is given by
the dominant Eigen values of A1 the stable age/stage
distribution by corresponding Eigen vector. Rate of
decay or death rate was designated as –μ and
calculated at each time interval.
2.5 General mathematical formulation of the
population dynamics Model:
As the number of individuals changes with time t,
=.
=.
∫
=.∫
Log N = m.t + C
at t = 0, N=N0
Since log N0 = C
Log N = mt + log N0
0
=
So,
N = N0 emt
Put emt = λ = growth rate
N= λ N0
or N(t+1) = λN(t)
where t = the initial observation year
t +1 = a time period of one year after the initial
observations were recorded
When λ equals 1, the population size is constant but
increases or decreases when λ is more than 1 or less
than 1 respectively.
2.6 Application of the Model
The age distribution vector were derived from log
graphs where the abscissae demarcated the length of
span for each age group scaled in such a way that the
length of life span for seedling stage equals one unit
and on the ordinates the density of different age
groups were plotted.
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The contribution to the population at time t+1 can be
obtained by multiplying Nt by the following matrix
comprised of a×a as submatrices,
where, AA to AD represent the survivorship
of 4 stages during the time interval.
AA
AB
AC
AD
2.7 Groups and Cohorts
In weed population 4 stage classes were recognized
which have specific longevity. Taking seedling state
life span as the unit, all the other 3 stage classes were
subdivided into a number of cohorts for projecting
population structure of future time as it was assumed
that the individuals in a particular ontogenetic stage
with the experimentally derived longevity can be
splitted over cohorts. Thus, taking the unit time
interval of 5 days in case of Euphorbia hirta the total
life span of the weed can be splitted into 17 cohorts.
Thus, the column vector N (t) comprised of age
classes (derived graphically on log scale.
2.8 Survival Coefficients
Survival coefficients were calculated for all stage
classes and are regarded as the proportion of
individuals borne at a given time, actually the
survival schedule of the individual takes into
consideration the number of individuals surviving to
a particular stage. To obtain survival coefficients, it
was seen that how many individuals borne at a
particular time survived the first interval of time,
how many the second, how many the third and so on
until no more were alive. Thus probabilities of
surviving from one age group to the next were
calculated. The population structure at t+1 time can
be derived by multiplying N(t) with a block diagonal
matrix with diagonal submatrices AA, AB, AC and
AD where AA to AD indicated the survivorship of
four stage classes during the time interval through
17 age groups in Euphorbia hirta. The number of
groups in seedling, juvenile, young and mature stage
classes were 1,2,5 and 9 for E. hirta.
Based on age transitions the matrices were
computed and equal to the proportion of plants that
were in the jth age class at time t that entered the ith
age class at time t+1.
2.9 Possibility of Seed Bank
The area had pure standing populations of the weed
studied. The rate of migration and entrance
(immigration) of propagules were assumed to be
constant taking into consideration the fact that
propagules had equal chance of dispersal in the field.
The model was thus simplified as effect of wind
velocity etc. on dispersal was not seen. The effect of
the dispersal agencies if any, was supposed to be
counterbalanced as the studied areas were located
within the pure stands of the weed population.
2.10 Projecting Population Structure
Leslie Matrix [56] modified after Gupta [71]
describing the contribution of each age group to
every other group during the time interval (t, t+1)
was employed with a column vector N(t) including
the number of individuals in each group. Mostly
emergent seedling population densities for 2021
were utilized as initial values in conventional Leslie
Matrix form for simulation, the projected population
structure and number of individuals in different age
group was compared with the observed data in the
field for the whole year. Relationship between
seedling emergence and edaphic variables was
regressed.
3 RESULTS AND DISCUSSION
3.1 Patterns of Growth
With experimental results available on E hirta, it
was found that the time of onset of flowering was 16
days which continued for 25 days, the maximum
fruiting period was 45 days, the limit age of
individuals involved in reproduction was 41 days
and the life expectancy was 11 weeks.
3.2 Fluctuations in Population
Dynamics of E. hirta population for 2021 and
2022 are reflected in Fig 1.In E hirta, the fluctuation
range for seedling population varied from 47.3% to
37.5% in 2021 whereas54.36% to 41.97%in
2022,for juvenile population, the range was36.25 %-
26.31%in 2021 and 36.56% to 26.1%in 2022for
flowering adult population, the range was 23.45%
to16.79%in 2021 24.13%-15.15 %in 2022,for
flowering-fruiting adult, population range was from
7.89%-1.92 % in 2021 to 3.70%-1.94 % in 2022
respectively(Fig1).
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Fig.1: Dynamics of Euphorbia hirta population for
(A) 2021 and (B) 2022((Density-Individuals
0.25 m-2).
3.3 Recruitment, Nutrient status and seedling
emergent population
The outbreak in seedling population was obtained in
December-January months in E. hirta.
An invasive species establishes when repeated
reproduction and survival of individuals result in a
population capable of maintaining itself in the wild
[72]. Survival and reproduction rely on many
abiotic (nonliving) factors that can either promote
or hinder invasive species establishment.
The multiple linear regression derived was
Y=-221.07+1.69 X1-16.51 X2+189.5 X3+820.73
X4-5.67X5 +0.59 X6
Where Y = seedling establishment
Numericals - 221.07 is the constant whereas other
numericals are Regression coefficients.
X1- Soil Moisture, X2-pH, X3-Soil Organic
Carbon, X4- Total Nitrogen, X5- Available
Phosphorus and X6- Exchangeable Potassium
The multiple linear regression in E. hirta showed
that the species responses were linear to the abiotic
factors studied on seedling establishment and were
significant both at 0.05 and 0.01 levels. (r2=0.970,
F=26.948 at df 6, 5).
The relative maximum percentage contribution on
seedling establishment was exhibited by
Exchangeable Potassium (59%) followed by pH
(39.3%) in E. hirta.
3.4 Age Structure and Age Pyramids
The age structure into 4 age groups viz. seedling,
juvenile, flowering adult and flowering –fruiting
adult was expressed as a combination of total annual
density in the studied area of corresponding age
group resulting in age pyramid. The fluctuations of
population age structure in the two years revealed
the fate of various cohorts in E. hirta (Fig 2).
.
Fig 2 Age pyramid for populations of Euphorbia
hirta
3.5 Age Specific Survival and Mortality
The monthly survival and mortality percentages for
seedling and juvenile age group of weed population
for a period of 2 years were computed. During the
study period, the maximum survival percentage by
seedling population was exhibited in the month of
June (96%) and July (95.56%) in Euphorbia hirta
for the year 2021 and 2022 respectively. The
maximum mortality % in seedling stage was
exhibited in the months of April (55.56%) and
September (46.67%) in the species for the year 2021
and 2022.
Whereas the population in juvenile stage showed the
maximum survival percentage in the months of June
(86.36% and 77.77%) for the year 2021 and 2022
respectively. The maximum mortality percentage in
the months of May (64.04% and 60.53%) for 2021
and 2022 respectively.
0
20
40
60
Density
J F M A M J JU A S O N D
Fruiting
Flowering
Juvenile
Seedling
Months
(A)
Fruiting Flowering Juvenile Seedling
0
20
40
60
Density
J F M A M J JU A S O N D
Fruiting
Flowering
Juvenile
Seedling
Months
(B)
Fruiting Flowering Juvenile Seedling
0
200
400
600
Density
Seedling Juvenile Flowering Fruiting
T1
T2
(D)
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3.6 Survival Coefficients
The survival coefficients showed a flux in various
age groups within the same species. Fig 3 reveals the
survival coefficients for different months on the
basis of average of two years value in 4 age groups
of weed population in Manipur. They are the
possible transitional probabilities and reflected the
temporal variation in population structure.
Fig 3 Survival coefficients for Euphorbia hirta
L.for four different age groups
The lower value of survival coefficients was
obtained for seedling population in Euphorbia hirta
in September that revealed the risky nature of this
stage in the life span.
3.7 Growth Rate (λ)
The weed population of Eurphorbia hirta exhibited
an annual percentage gain of 10.55%. The number
of new individuals in 2 years (2021 and 2022)
revealed the fate of various cohorts and subcohorts
in the weed species. The average survival and
mortality percentages were 75.65% and 24.35%
respectively hirta exhibited positive density
dependent correlation, growth rate exceeded value
1.0 in the species, the annual decay rate was
0.235(Fig 4).
Fig 4 Growth Rate (λ) and Decay Rate (–μ) for the
populations of Euphorbia hirta
3.8 Population behavior
Gupta (71;73;74;75;76) made attempts to combine
age and structure stage into one and analysed the
demography of herbaceous annuals by matrix
model.
The Leslie matrix model take the general form of
N(t+1) = A.N (t),
where, N(t+1) and N(t) are vectors
representing age or stage class distribution of
individuals at time t+1 and t, A is the matrix defining
the age or stage specific survival and fecundity
values. The rate of emigration and immigration of
propagules were assumed to be equal. The model
was thus simplified. The asymptotic population
growth rate λ is given by the dominant Eigen values
of A1 the stable age/stage distribution by
corresponding Eigen vector. Rate of decay or death
rate was designated as (–μ and calculated at each
time interval.
The age distribution vector was derived from log
graphs where the abscissae demarcated the length of
span for each age group scaled in such a way that the
length of life span for seedling stage equals one unit
and on the ordinates the density of different age
groups were plotted in all the herbaceous annuals
populations. It was assumed that individuals in a
particular ontogeny with experimentally derived
longevity can be splitted into subcohorts
The Column Vector N(t) comprised of: -
N(t) = (68.0, 56.0, 43.0, 35.0, 27.2, 21.0, 14.1, 8.0,
7.0, 5.3, 5.0, 4.0, 3.8, 3.6, 3.2, 3.0,2.8).
In E hirta through 17 cohorts (with Seedlings life
span 5 days)
The contribution to the population at time t+1 can be
obtained by multiplying Nt by the matrix defining
the age or stage specific survival matrix. For
simulations the monthly emergent seedling
population for the initial year were analysed in case
of all plants and multiplied with survivorship co-
efficient.
3.9 Population Projection
The age structure of projected population for 2022
revealed its resemblance with observed age
structure for 2022 population in field conditions. It
indicated that the projection matrix is satisfactory in
differentiating thee age specific characteristics of
cohorts. In both the cases of projected and observed
populations, the largest category of individuals was
composed of seedlings followed by juveniles, then
flowering stage and lastly fruiting adult stage plants.
Thus, a remarkable similarity between projected and
observed age group distribution curves were noticed
(Fig.5).
JFMAMJJASOND
0.0
2.0
4.0 Fruiting Flowering
Juvenile Seedling
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
J F M A M J J A S O N D
Months
Growth rate
-0.02
0.03
0.08
0.13
0.18
0.23
0.28
0.33
Decay rate
Growth Rate (λ1) Growth Rate (λe)
Decay Rate (-μ1) Decay Rate (-μ2)
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Fig 5 Projection for Euphorbia hirta L Population
(Leslie Matrix Model)
The simulation indicated a striking similarity
between observed and projected populations. It
indicated that the projection matrix was satisfactory
in differentiating the age specific characteristics of
cohorts and sub-cohorts. The present study also
substantiated the earlier works [71;73;74;75;76].
The matrix model provided a path elucidating the
dynamics of population in which reproduction and
survival coefficient for seedling stage suggested it a
critical stage in life cycle of E. hirta. Thus, the lower
value of survival coefficients obtained for seedling
population in the month of September revealed
that the mortality risk was highest at seedling stage
suggesting to have the control measures at seedling
stage In, Portulaca oleracea [75;76], the seedling
stage was also regarded as risky due to minimum
survival coefficient value. Earlier works [71;73] on
Parthenium hysterophorus revealed flowering stage
a critical stage in the life cycle of this notorious weed
that approved that the effective control of the weed
may be brought about by regulating it at flowering
stage.
Similarly, for Tridax procumbens, Linn., flowering
was obtained as a sensitive stage. Thus, it appears
that a division of resources between vegetative and
reproductive organ is important for it [74;75],
whereas in Bidens pilosa survival co-efficient of
juvenile stage was found to be low indicating that
competition for nutrient and space (Dense turf) is an
important factor in regulating it [74;75].
The study reveals that seedling stage is one of the
riskiest phases in the life-history of E. hirta. It is
considered to be the most vulnerable stage in the life
of the plant [77]. Fenner & Thompson [ 78] owe this
to reduction in biomass, if even reduced in small
amount may lead to the death of the plant. Seedlings
face threats to their establishment from natural
enemies to resource limitations and insufficiencies
in sites suitability [79]. Seedling recruitment can
depend on abiotic and/or biotic variation at very
small scales (e.g. meters or weeks), yet be a major
demographic driver of community dynamics and
species distributions [80;81] Burkey and Stenseth
[82] demonstrated with a seasonal model how the
value of the resource to each individual may be
reduced due to its patchy distribution. It was
concluded that the difference among population of
various weed species depend on the difference of
their transition probabilities of matrices in the same
year and on the difference in stochastic processes.
As also observed by Aberg [83], the difference was
reflected in value of λ and –μ rates.
4 Conclusion
Matrix population models offer a tool for
identifying the demographic processes that have the
greatest impact on population growth rates in order
to better understand population dynamics and
potential management strategies for invasive plant
species. I demonstrate that the population under
study has a higher growth rate and is
demographically stable through the examination of
asymptotic growth rate and demographic
parameters. Population declines of invasive species
may be achieved by focusing control on
demographic processes (survival) to target for
reductions in population. Reducing the survival and
growth of the species would have the biggest impact
on lowering the overall population growth rate.
Invasive species can experience population losses
by concentrating management on demographic
processes. Applying density thresholds to a
transient invasive species (E.hirta), I examined how
population density affected population projection
and offered for management recommendations.
As invasive species management is concerned with
maximizing mortality and lowering the reproduction
and minimizing the loss of resources resulting from
invasive species competition with crops, it is
suggested to target the population for management
measures at seedling stage so as to eliminate the
probability of reaching the adult stage thereby
setting seeds for further infestation and spread. Our
results provide a first evaluation of the demography
of E. hirta, a species of economic concern, and
J F M A M J J A S O N D
0
10
20
30
40
50
60
70
Months
Stages of plant species
(Projected) Seedling
Juvenile
Flowering
Fruiting
(Observed) Seedling
Juvenile
Flowering
International Journal of Applied Sciences & Development
DOI: 10.37394/232029.2024.3.12
Asha Gupta
E-ISSN: 2945-0454
137
Volume 3, 2024
provide the first structured population model of a
representative of the Euphorbiaceae family, thus
contributing to our global knowledge on plant
population dynamics.
5 Recommendations and Future work
Invasive species, a global threat due to climate
change and human disturbances, require urgent
management strategies. Early detection and swift
action are crucial for eradicating these species which
disrupt ecosystems, impact adversely biodiversity,
introduce diseases, and cause financial burdens.
Matrix models provide the tools that can identify
high-risk stage in the life cycle, prioritizing
management and population eradication.
Recruitment from seed banks impacts population
structure in time and space, with age structure
resulting from long-term persistence in the seed
bank. Changes in age structure and population size
significantly influence demography, necessitating
inclusion of age-structured seed bank dynamics in
demographic models that I intend to include in my
future work.
Acknowledgement
The author wholeheartedly thanks the three
anonymous reviewers whose criticisms and
constructive suggestions helped to a great extent in
improvement of the present paper. Author also
acknowledges the Head, Centre of Advanced Study,
Department of Life Sciences, Manipur University
for the facilities.
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