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F
ree-roaming cats represent challenges to the veteri-
nary profession and society. Free-roaming cats con-
tribute to a public health hazard through the risk of
transmitting rabies and other zoonotic agents.
1-6
These
cats may be infected with a variety of feline pathogens
and function as a reservoir of infection for owned
cats.
1,2,7,8
The welfare of free-roaming cats concerns
society because they are frequent victims of vehicular
collisions and fights between themselves and other ani-
mals.
1,3,5,6,9,10
Cats also are efficient predators, and results
of numerous studies
2,3,5,11,12
indicate their detrimental
impacts on native wildlife. For example, the British
population of 9 million cats has been estimated to be
responsible for the deaths of 57 million mammals, 27
million birds, and 5 million reptiles and amphibians
per year
11
; because the US cat population has been esti-
mated at 100 million,
12
the potential impact of domes-
tic cats on American wildlife is high. There also are
emotional and financial costs to society caused by con-
cern for these cats, their prey, and the attempts to mit-
igate the perceived and real damages caused by unman-
aged populations of free-roaming cats.
6,9,10,13
Despite
recognition of the problems, there is no consensus on
how such populations should be managed.
Two management schemes predominate. Traditional
animal control plans manage free-roaming cats through
capture and removal. Whereas some captured cats are
returned to their owners or adopted as pets, most are
euthanatized as unwanted, excess cats. Traditional ani-
mal control is often constrained by resources and rarely
sustains active cat population management on a broad
scale. A more recently proposed management alternative
is to maintain stable, nonbreeding populations. These
programs are founded on capturing, neutering, and
releasing cats with additional management processes of
disease testing, vaccination, feeding, adoptions, and
monitoring being components of some programs.
Although trap-neuter-return (TNR) programs are pre-
sented as an alternative to euthanasia and an effective
means of population management,
1,4,6,10,14,15
these pro-
grams have not been developed on a large scale, limiting
their assessment as a tool for decreasing cat populations.
Further, TNR programs are often instituted because of
society’s aversion to euthanasia as a method of control.
Thus, much of the debate surrounding the use of
euthanasia or TNR has a substantial emotional compo-
nent. Therefore, an objective evaluation of TNR and
alternative strategies for managing cat populations would
make a valuable contribution toward sound manage-
ment.
Matrix population models are a primary tool used
in wildlife management to set annual guidelines on
hunting, trapping, and fishing; explore population
dynamics; and develop management plans for endan-
gered species.
16-29
These models use estimates of age- or
stage-specific vital rates (reproduction and survival) to
project future population structure. Analysis of these
models allows objective comparison of the efficacy of
different management actions and permits identification
of key factors influencing population dynamics.
18,29,30
For our study, published data from studies of urban,
free-roaming cat populations were used to parameterize
JAVMA, Vol 225, No. 12, December 15, 2004 Scientific Reports: Original Study 1871
SMALL ANIMALS
Use of matrix population models
to estimate the efficacy of euthanasia
versus trap-neuter-return for management
of free-roaming cats
Mark C. Andersen, PhD; Brent J. Martin, DVM, DACLAM; Gary W. Roemer, PhD
Objective—To evaluate the efficacy of trap-neuter-
return and trap-euthanatize management strategies
for controlling urban free-roaming cat populations by
use of matrix population models.
Design—Prospective study.
Sample Population—Estimates of free-roaming cat
populations in urban environments.
Procedure—Data from the literature describing the
biology of free-roaming cat populations in urban envi-
ronments were gathered. A matrix population model
was developed with a range of high and low survival
and fecundity values and all combinations of those
values. The response of population growth rate to a
range of management actions was assessed with an
elasticity analysis.
Results—All possible combinations of survival and
fecundity values of free-roaming cats led to predic-
tions of rapid, exponential population growth. The
model predicted effective cat population control by
use of annual euthanasia of ≥ 50% of the population
or by annual neutering of > 75% of the fertile popula-
tion. Elasticity analyses revealed that the modeled
population was most susceptible to control through
euthanasia.
Conclusions and Clinical Relevance—Free-roaming
cat populations have a high intrinsic growth rate, and
euthanasia is estimated to be more effective at reduc-
ing cat populations than trap-neuter-return programs.
(
J Am Vet Med Assoc
2004;225:1871–1876)
From the Department of Fishery and Wildlife Sciences, New Mexico
State University, Las Cruces, NM 88003-0003 (Andersen,
Roemer); and the Division of Laboratory Animal Medicine,
Medical College of Ohio, Toledo, OH 43614-5806 (Martin).
Partially funded by the New Mexico Agricultural Experiment
Station.
The authors thank D. Kallakuri for assistance with the matrix popu-
lation projection model.
Address correspondence to Dr. Andersen.
04-01-0021.qxd 11/23/2004 1:23 PM Page 1871
a matrix population model
17
and explore how cat popu-
lations may respond to various forms of control. The pri-
mary objective was to compare the efficacy of TNR pro-
grams versus euthanasia programs as methods of cat
population management. Because TNR programs affect
reproduction and euthanasia programs affect survival,
these 2 approaches should be fundamentally different
with respect to their impact on the growth rate of cat
populations. To address this hypothesis, an elasticity
analysis of the matrix population models was conducted
to explore how alterations in the estimates of stage-spe-
cific vital rates would influence population growth rate.
Elasticity analysis is a form of perturbation analysis that
allows determination of the rate of change of population
growth rate in response to changes in individual vital
rates.
17
This allows vital rates to be ranked in order of
their influence on population growth.
Materials and Methods
Model description and construction—The matrix popu-
lation model
31,32
allows classification of the population either by
age or stage classes (eg, larva, juvenile, adult). The model
advances a given population structure ahead through 1 time
interval to a new projected population structure. Through that
time interval, reproduction and survival occur within each age
or stage class in the population at rates specific to that age or
stage class. For instance, young animals, such as juveniles, have
a probability of surviving (usually < 1) to the reproductive
adult stage, thereby increasing the numbers in the adult stage
and decreasing the numbers in the juvenile stage. Similarly,
reproduction by different stages at time t would contribute to
the number of young present at time t + 1. Thus, the survival
probabilities and reproductive rates of each age or stage class
would contribute to a new population structure at time t + 1.
Under this model, changes in population structure are
denoted by the equation n(t + 1) = An(t), where n represents a
vector of age or stage classes describing the population struc-
ture at times t + 1 and t, respectively, and the population pro-
jection matrix A contains the survival probabilities and repro-
ductive rates acting on the population through each time inter-
val. The dominant eigenvalue of the population projection
matrix, denoted by λ, is the intrinsic or asymptotic growth rate
of the population. If λ = 1 there is no net change in the popu-
lation size. Values > 1 mean that the population is increasing;
values < 1 mean that the population is decreasing.
17
Model parameter estimation—Vital rate data were gath-
ered from published studies of free-roaming cats in urban envi-
ronments; preference was given to studies of unmanipulated
populations. Given the variability of the vital rate data, low and
high extremes of parameter estimates and all combinations of
those extremes were used in the analysis. The available data
were insufficient for an age-structured classification but
amenable to a stage-structured model with 2 stages: individuals
≤ 1 year old, called juveniles, and individuals > 1 year old, called
adults. This classification of the population model conformed to
the level of detail of the management actions that were evaluat-
ed. The model time step was 1 year. The model considered only
the female population; this is appropriate if the population is not
mate-limited or if the vital rates of the 2 sexes are identical.
17
Reproduction in cats is relatively well documented, and
values from the literature were used to develop the model.
Fetus number has been reported as mean ± SD of 3.6 ± 0.2
kittens/dam in a program to spay free-roaming cats.
15
This can
be conservatively considered a typical litter size and is com-
parable to values found in other studies.
2,9
Mean number of
litters per female per year has been reported to be from 1.1 to
2.1.
10,15
Puberty in female cats varies with time of year of their
birth, but first conception has been reported to be at a mean
of 212 days of age.
2
Sex ratios are consistently near 50:50.
15
Based on these values, the fecundity of adult female cats
was estimated as a product of kittens per litter, litters per
year, and sex ratio at birth. Thus, our low estimate of fecun-
dity was 3.6 X 1.1 X 0.5 = 1.98 female offspring/y; our corre-
sponding high estimate was 3.6 X 2.1 X 0.5 = 3.78 female off-
spring/y. Juvenile females have reduced mean fecundity
because most are prepubertal. This reduction is equal to the
factor (365 – 212)/365, which yielded 0.83 female offspring/y
to 1.58 female offspring/y for the low and high estimates of
juvenile fecundity, respectively.
Survival probabilities have not been as thoroughly docu-
mented. In a long-term study,
2
annual juvenile survival was
reported as approximately 75%. In another report,
33
it was sug-
gested that annual survival is closer to 50%. Published figures
for survival rates of adults include 33% survival over a 42-month
period
34
and 67% survival over an 18-month period.
4
Anecdotal
reports estimate adult life span of feral cats at 2 to 3 years.
Survival rates were computed under the assumption that
the juvenile and adult classes reported in the various studies
were consistent with this model’s juvenile and adult classes.
The low estimate of survival for juveniles was therefore 0.5, and
the high estimate was 0.75. Reported survival rates for adults
were determined over periods longer than 1 year; annual sur-
vival rates were estimated by use of the geometric probability
distribution as an approximate discrete lifetime distribution.
35
This method assumes that survival probabilities remain con-
stant over the entire period for which they are being estimated.
This assumption may not be true if there are age-dependent
effects on survival. For example, if younger adults learn as they
age, survival may be enhanced with time, whereas senescence
may cause the opposite effect in older adults, reducing survival
over time. It is likely that both processes occur, but the avail-
able data did not allow us to address this issue. The distribution
function for the geometric distribution was P = 1 – S
t
, where S
is the per-time-unit survival rate, and P is the probability that
an individual will die by time t. Thus, the expression for S given
t and P was S = exp(ln[1 – P]/t).
This expression was used to obtain 4 estimates of annual
survival rates for the adult class based on 3 values of P.
Calculations based on a 2-year and 3-year life span, assuming
that P = 0.50, yielded annual survival rates of 0.707 and 0.794,
respectively. A monthly survival rate of 0.974 was calculated
from a survival probability of 0.33 over 42 months (ie, P = 0.67
and n = 42), and this monthly rate was converted to an annual
rate of 0.729 (ie, 0.974 to the 12th power). Similarly, a month-
ly survival rate of 0.978 results from a survival probability of
0.67 over 18 months (P = 0.33 and n = 18). This monthly rate
converts to an annual survival rate of 0.766 (0.978 to the 12th
power). Thus, by use of 4 reported estimates of adult survival,
the estimated annual rate of adult survival varied from 0.707 to
0.794. Therefore, the low and high values for adult survival
rates used in the model were 0.7 and 0.8, respectively.
Although breeding is skewed towards the first half of
the year and only a few litters are born during the months
of October through December, cats essentially can breed
throughout the year.
2,9,15
Continuously breeding populations
are most easily modeled by assuming that all births take
place at the midpoint of the time interval.
17
Under this
assumption, parental individuals must survive to that mid-
point and the resultant offspring must survive through the
remaining half of the time interval. Half-year survival rates
are equal to the square root of the annual rate. Thus, the
matrix elements are as follows:
1872 Scientific Reports: Original Study JAVMA, Vol 225, No. 12, December 15, 2004
SMALL ANIMALS
[
S
0
•
F
0
√
S
0
•
F
1
•
√
S
1
]
S
0
S
1
04-01-0021.qxd 11/23/2004 1:23 PM Page 1872
where S
0
and S
1
are annual rates of survival for the juvenile and
adult stage classes, and F
0
and F
1
are the estimates of fecundity
for those same stage classes. The composite matrix elements in
the top row of the matrix are equal to the stage-specific repro-
ductive rates (R
0
and R
1
) for juveniles and adults, respectively.
Model analyses—Analyses were performed with soft-
ware written in a standard programming language.
a
Intrinsic
rates of increase (λ) were calculated with all combinations of
high and low fecundity and survival rates. Fecundity was
reduced by 10%, 25%, 50%, and 75% to simulate TNR pro-
grams of increasing rigor. Trap-euthanatize programs were
modeled through several combinations of reductions of juve-
nile and adult survival by 10%, 25%, 50%, and 75%.
The geometric mean of λ, computed across all combina-
tions of fecundity and survival rates, was used as a summary
statistic in assessment of population management strategies.
Elasticity analysis was used to provide a broader view of the
demographic basis of the effects of the different management
strategies. All elasticity values were calculated with a com-
puter program.
30,a
Uncertainty in vital rate estimates may influence the
outcome of computations of the elasticity values. These pos-
sible effects may be accounted for by computing elasticity
values for a large number of simulated population projection
matrices, in which the vital rate values are randomly chosen
to lie between predetermined limits. The upper and lower
bounds chosen for the vital rates were the high and low esti-
mates, respectively, for each rate.
Results
The high and low vital rate values resulted in 8 pos-
sible matrices that each yielded λ > 1 (Table 1).
Population growth rate ranged from a high of 2.49 for the
highest estimates of fecundity and survival for both juve-
nile and adult classes to a low of 1.34 for the lowest com-
binations. All remaining combinations of parameters led
to intermediate intrinsic rates of population increase. The
geometric mean of these 8 baseline values of λ was 1.84.
The geometric mean intrinsic rates of increase for
the matrices simulating a TNR (ie, with reduced fecun-
dity values), even for quite large reductions in fecundity,
were still > 1 (Table 2). A 75% reduction in fecundity for
all reproductive females (corresponding to ongoing
spaying 75% of the female population) yielded λ = 1.08.
The geometric mean intrinsic rates of increase for
the matrices simulating a euthanasia program (ie, with
reduced survival values) revealed that reductions of ≤
25% were not sufficient to lead to predictions of declin-
ing cat populations (Table 2). However, reductions in
both juvenile and adult survival by ≥ 50% yielded a geo-
metric mean intrinsic rate of increase < 1. A 75% reduc-
tion in both adult and juvenile survival led to λ = 0.47,
meaning that a cat population subjected to such a pro-
gram would be approximately halved every year.
In nearly all scenarios, λ was more sensitive to
changes in survival than fecundity (Figure 1). A 25%
JAVMA, Vol 225, No. 12, December 15, 2004 Scientific Reports: Original Study 1873
SMALL ANIMALS
Juvenile Adult
Scenario Fecundity survival survival Matrix
λλ
1 High High High 1.189 2.928 2.49
0.75 0.8
2 High High Low 1.189 2.739 2.40
0.75 0.7
3 High Low High 0.792 2.391 1.89
0.5 0.8
4 High Low Low 0.792 2.236 1.80
0.5 0.7
5 Low High High 0.622 1.534 1.79
0.75 0.8
6 Low High Low 0.622 1.435 1.70
0.75 0.7
7 Low Low High 0.83 1.252 1.61
0.5 0.8
8 Low Low Low 0.415 1.171 1.34
0.5 0.7
For juvenile fecundity, low = 0.83 female offspring/female per
year, high = 1.58 female offspring/female per year; for adult fecundi-
ty, low = 1.98 female offspring/female per year, high = 3.78 female off-
spring/female per year. For juvenile survival, low = 0.5, high = 0.75; for
adult survival, low = 0.7, high = 0.8.
For each matrix, upper left value represents juvenile reproductive
rate, upper right value represents adult reproductive rate, lower left
value represents juvenile survival, and lower right value represents
adult survival.
Table 1—Baseline population projection matrices used in simul-
ations of free-roaming cat populations and their intrinsic rates of
increase (λ). Each scenario corresponds to a different estimate
of fecundity and survival.
Figure 1—Elasticity of the intrinsic rate of population increase in
response to changes in survival and fecundity rates of free-
roaming cats under 8 scenarios corresponding to the matrices in
Table 1. Survival and fecundity rates that yield larger elasticity
values are expected to have a greater influence on the intrinsic
rate of population increase. F
0
= Juvenile fecundity. F
1
= Adult
fecundity. S
0
= Juvenile survival. S
1
= Adult survival.
Fecundity Juvenile survival (%) Adult survival (%)
λλ
Baseline Baseline Baseline 1.84
10 NC NC 1.73
25 NC NC 1.59
50 NC NC 1.35
75 NC NC 1.08
NC 10 10 1.63
NC 10 25 1.53
NC 25 10 1.46
NC 25 25 1.36
NC 50 50 0.91
NC 50 75 0.73
NC 75 50 0.62
NC 75 75 0.47
NC = No change in the vital rate from the baseline values.
Table 2—Geometric mean λ for free-roaming cat populations
with either reduced fecundity, simulating the effects of a trap-
neuter-return program, or reduced survival, simulating the
effects of a euthanasia control program. Geometric means were
calculated over the 8 baseline matrix models in Table 1 with
fecundity or survival reduced by the percentage indicated.
[
[
[
[
[
[
[
[
]
]
]
]
]
]
]
]
04-01-0021.qxd 11/23/2004 1:23 PM Page 1873
reduction in fecundity of the cat population led to a
growth rate of 1.59, whereas a 25% reduction in sur-
vival reduced growth rate to 1.36 (Table 2). Fifty per-
cent and 75% reductions in survival similarly led to
greater reductions in population growth than did the
50% and 75% reductions in fecundity. Elasticity values
ranged from as high as 0.73 for juvenile survival in 1
scenario to as low as 0.11 for juvenile fecundity in
another scenario. Elasticity for the survival rates
ranged from 0.27 to 0.73, whereas elasticity for the
fecundities remained < 0.30. This result was mirrored
by the stochastic analyses of λ (Table 3). Elasticity val-
ues for the mean vital rates ranged from 0.24 to 0.76,
and the mean elasticity values for the random matrices
ranged from 0.20 to 0.65. The highest elasticity values
were for juvenile survival, suggesting that population
growth is most sensitive to this vital rate, followed by
adult survival. Control strategies that target survival of
free-roaming cats should be more effective at reducing
cat populations than those that target fecundity.
Discussion
Matrix population models have been used success-
fully in many population management strate-
gies.
18,20,22,23,29,36
Such models have been used to assess
the viability of populations of endangered plants
19,26
and animals,
21,22
to assess the impacts of wildfires
24
and
pollutants on natural populations,
25,27,28
and to study
the pathogenicity of an external parasite.
37
Matrix mod-
els are often preferred in management applications
because of the degree of development of the underly-
ing mathematics, the level of realism of the models,
and the ease of parameter estimation.
38
The geometric mean growth rate rather than the
arithmetic mean is the appropriate measure of the most
likely growth rate of a stochastic population growth
process. The geometric mean best represents the
expected rate of growth of a population in which one
of the possible population matrices is randomly chosen
at each time step. In other words, if one of the different
possible population projection matrices considered
was randomly selected and a population was to grow
according to that matrix for that time step, and anoth-
er matrix was randomly selected for the following time
step, then over time, the population’s long-term growth
rate would be the geometric mean, not the arithmetic
mean, of the set of possible growth rates.
17,30
Thus, the
use of the geometric mean of λ as the summary mea-
sure directly and explicitly recognizes the uncertainty
in the parameter estimates.
Demographic elasticity analysis is a way of calcu-
lating the effect of small changes in the vital rates of a
population on the population’s rate of growth.
Analytically, this can be computed as the partial deriv-
ative of λ with respect to each individual vital rate,
holding the others constant. These partial derivatives
are referred to as sensitivity values; when these values
are multiplied by the ratio of the vital rate in question
to λ (to scale for differences in the vital rates them-
selves), they are referred to as elasticity values.
17
The
elasticity of λ can be calculated with respect to partic-
ular vital rates or with respect to the matrix elements
themselves. In this analysis, the elasticity of λ was cal-
culated with respect to each vital rate (ie, stage-specif-
ic fecundity and survival) because the matrix element
for fertility was a composite of several vital rates.
Examination of elasticity values is valuable in
assessing management strategies for free-ranging pop-
ulations. Such strategies nearly always have age-specif-
ic effects, representing a perturbation in a particular
vital rate, and are aimed at either increasing or reduc-
ing the target population’s growth rate. Management
strategies intended to alter vital rates with particularly
high elasticity values are more likely to achieve their
goal than strategies that target vital rates with low elas-
ticity.
30
Management programs for free-roaming cat popu-
lations typically focus on either survival (euthanasia
programs) or fecundity (TNR programs). Because
these 2 approaches target different vital rates, they may
have fundamentally different outcomes with respect to
their influence on cat population growth rate. Under
scenarios lacking control, feral cat populations were
predicted to grow rapidly because all values of λ are
substantially > 1. The model results further suggested
that a reduction in survival might have a more pro-
found effect on cat population growth rate than a
reduction in fecundity. A 50% reduction in annual sur-
vival rate was predicted to result in a cat population
that declined by approximately 10% per annum,
whereas a large reduction in annual fecundity (75%) in
both the juvenile and adult stages was predicted to
result in an increasing population. The interpretation
that survival had a greater predicted influence on pop-
ulation growth rate also was supported by results of the
elasticity analyses; cat population growth rate was
more sensitive to survival regardless of the control sce-
nario.
Juvenile survival has been identified as a key pop-
ulation management target in matrix-modeling studies
of other species.
18,21
Given the uncertainty of the vital
rate estimates, the sensitivity of urban cat populations
to changes in adult and juvenile survival cannot be
confidently distinguished. Nevertheless, changes in
survival are always predicted to have a greater influ-
ence on population growth than changes in fecundity.
Matrix population models have also been applied
to owned populations of pet dogs and cats.
39-41
An age-
structured matrix of pet cats yielded a λ of 1.21 for cats
through the first 5 years of life.
39
This value is lower
than any growth rate calculated for the unmanaged
population structures used here; however, those
authors were modeling pet populations in which
responsible owners had a substantial number of cats
neutered or prevented breeding by fertile cats. A spay
1874 Scientific Reports: Original Study JAVMA, Vol 225, No. 12, December 15, 2004
SMALL ANIMALS
Table 3—Elasticity of λ in free-roaming cat populations for mean
vital rates and random matrices derived from a range of values
for each vital rate.
Elasticity of random matrices
Elasticity of
λλ
for
Vital rate mean vital rates Minimum Maximum Mean (SD)
Juvenile fecundity 0.34 0.12 0.33 0.20 (0.045)
Adult fecundity 0.28 0.26 0.33 0.30 (0.016)
Juvenile survival 0.76 0.55 0.72 0.65 (0.034)
Adult survival 0.24 0.28 0.45 0.36 (0.035)
04-01-0021.qxd 11/23/2004 1:23 PM Page 1874
rate of 88% would be needed to stabilize population
growth if all fertile cats were free to breed.
39
In a sepa-
rate study,
41
a growth rate of 1.02 was calculated for a
citywide pet cat population with a spay rate of 85.7%.
This result is similar to the most intensive fecundity
reduction used in this study, in which a 75% reduction
in fecundity yielded a geometric mean λ of 1.08. These
results suggest that management actions that reduce
fecundity in excess of 75% of the fertile population
would need to be maintained, on an ongoing basis, to
cause a population decrease in a TNR program. Thus,
TNR programs are not likely to convert increasing cat
populations into declining populations or even stable
populations until the neutering rate is quite high.
Nevertheless, population decreases under TNR
programs have been recorded. In 1 study,
4
a 26% pop-
ulation reduction over an 18-month period with an
approximately 70% neuter rate was reported. This
reduction also included a population reduction of 25%
through a concurrent adoption program. Another
study
10
revealed profound population reduction in a
managed cat population with essentially a 100% neuter
rate and adoptions occurring at a high rate (approx
47% of the population). Adoption programs are similar
in effect to euthanasia because these cats are perma-
nently removed from the free-roaming population.
Feral or free-roaming cat populations are subject
to additional population processes that were not con-
sidered here. The survival rates used may account for
extrinsic factors that would be expected to cause death
(eg, intraspecific aggression or disease), but the esti-
mates of fecundity used did not incorporate a measure
of density dependence, which might be expected to
lower reproductive rates at high population densities.
Density dependence was omitted for 2 reasons. First,
there are no reliable estimates of the reduction in
fecundity that would be expected at high population
densities; second, small populations would not be
influenced by density dependence to any great degree.
Emigration also is apt to be a substantial population
factor that was not considered. Emigration between cat
colonies has been reported,
1,4,10
and a substantial num-
ber of owned cats are reported to be adopted strays.
5,33,41
Evaluating the efficacy of euthanasia versus TNR pro-
grams would benefit from additional field studies that
estimate other population processes and from well-
designed monitoring programs run in parallel with
control programs.
a
MATLAB, The MathWorks Inc, Natrick, Mass.
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