Philippa Kaur

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Model descriptionThe system dynamics model divided an urban dog population into the following subpopulations: (i) free-roaming dogs (both owned and unowned free-roaming, i.e. unrestricted dogs found on streets), (ii) shelter dogs (unowned restricted dogs living in shelters), and (iii) owned dogs (owned home-dwelling restricted dogs) (Fig. 1). The subpopulations change in size by individuals flowing between the different subpopulations or from flows extrinsically modelled (i.e. flows from subpopulations not included in the systems model; the acquisition of dogs from breeders and friends to the owned dog population, and the immigration/emigration of dogs from other neighbourhoods).Ordinary differential equations were used to describe the dog population dynamics. The models were written in R version 3.6.128, and numerically solved using the Runge–Kutta fourth order integration scheme with a 0.01 step sizes using the package “deSolve”29,30. For the baseline model, Eqs. (1–3) were used to describe the rates of change of dog subpopulations in the absence of management.Baseline free-roaming dog population (S):$$frac{dS}{dt}={r}_{s}times Stimes left(1-frac{S}{{K}_{s}}right)+alpha times O-delta times S$$
(1)
In the baseline model, the free-roaming dog population (Eq. 1) increases through the free-roaming dog intrinsic growth rate (rs), and the abandonment and roaming of dogs from the owned dog population (α) and decreases through adoption to the owned dog population (δ). The intrinsic growth rate is the sum of the effects of births, deaths, immigration, and emigration, which are not modelled separately. In this model, the growth rate of the free-roaming dog population is reduced depending on the population size in relation to the carrying capacity, through the logistic equation (rreal = rmax(1 − S/Ks))31. In the baseline simulation, the free-roaming dog population rises over time, until it stabilises at an equilibrium size.Baseline shelter dog population (H):$$frac{dH}{dt}=gamma times O-beta times H- {mu }_{h}times H$$
(2)
The shelter dog population (Eq. 2) increases through relinquishment of owned dogs (γ) and decreases through the adoption of shelter dogs to the owned dog population (β), and through the shelter dog death rate (µh). There is no carrying capacity for the shelter dog population as we assumed that more housing would be created as the population increases. This allowed calculation of the resources required to house shelter dogs.Baseline owned dog population (O),$$frac{dO}{dt}={r}_{o} times Otimes (1-frac{O}{{K}_{o}})+beta times H+delta times S-alpha times O-gamma times O$$
(3)
The owned dog population (Eq. 3) increases through the owned dog growth rate (ro), adoption of shelter dogs (β), and adoption of free-roaming dogs (δ); and decreases through abandonment/roaming (α) and relinquishment (γ) of owned dogs to the shelter dog population. The growth rate of the owned dog population (ro) combines the birth, death, and acquisition rates from sources other than the street or shelters (e.g. breeders, friends) and was modelled as density dependent by the limit to growth logistic formula (1 − O/Ko).Parameter estimatesDetailed descriptions of parameter estimates are provided in the supplementary information. The simulated environment was based on the city of Lviv, Ukraine. This city has an area of 182 km2 and a human population size of 717,803. Parameters were estimated from literature, where possible, and converted to monthly rates (Table 1). Initial sizes of the dog populations were estimated for the baseline simulation, based on our previous research in Lviv32. The carrying capacity depends on the availability of resources (i.e. food, shelter, water, and human attitudes and behaviour33) and is challenging to estimate. We assumed the initial free-roaming and owned dog populations were at carrying capacity. Initial population sizes for simulations including interventions were determined by the equilibrium population sizes from the baseline simulation (i.e. the stable population size, the points at which the populations were no longer increasing/decreasing).Table 1 Parameter description, parameter value, and minimum and maximum values used in the sensitivity analysis for the systems model.Full size tableEstimating the rate at which owned dogs are abandoned is difficult, as abandonment rates are often reported per dog-owning lifetime32,34 and owners are likely to under-report abandonment of dogs. Similarly, it is challenging to estimate the rate that owned dogs move from restricted to unrestricted (i.e. free-roaming). For simplicity, we modelled a combined abandonment/roaming rate (α) of 0.003 per month, estimated based on our previous research in Lviv and from literature34,35,36. We derive the owned dog relinquishment rate (γ) from New et al.37. We estimated shelter (β) and free-roaming adoption rates (δ) from shelter data in Lviv. We set the maximum intrinsic growth rate for the free-roaming dogs (rs) at 0.03 per month, similar to that reported in literature17,19,38. We assumed that demand for dogs was met quickly through a supply of dogs from births, breeders and friends and set a higher growth rate for the owned dog population (ro) at 0.07 per month.We assumed shelters operated with a “no-kill” policy (i.e. dogs were not killed in shelters as part of population management) and included a shelter dog death rate (µh) of 0.008 per month to incorporate deaths due to euthanasia for behavioural problems or health problems, or natural mortality. We modelled neutered free-roaming dog death rate (µn) explicitly for the CNR intervention at a minimum death rate of 0.02 per month38,39,40,41.InterventionsSix intervention scenarios were modelled (Table 2): sheltering; culling; CNR; responsible ownership; combined CNR and responsible ownership; and combined CNR and sheltering, representing interventions feasibly applied and often reported27. Table 2 outlines the equations describing each intervention. To simulate a sheltering intervention, a proportion of the free-roaming dog population was removed and added to the shelter dog population at sheltering rate (σ). In culling interventions, a proportion of the free-roaming dog population was removed through culling (χ).Table 2 Description of intervention parameters and coverages for simulations applied at continuous and annual periodicities.Full size tableFree-roaming dog population with sheltering intervention:$$frac{dS}{dt}={r}_{s}times Stimes left(1-frac{S}{{K}_{s}}right)+alpha times O-delta times S-sigma times S$$
(4)
Shelter dog population with sheltering intervention:$$frac{dH}{dt}=gamma times O-beta times H- {mu }_{h}times H+sigma times S$$
(5)
Free-roaming dog population with a culling intervention:$$frac{dS}{dt}={r}_{s}times Stimes left(1-frac{S}{{K}_{s}}right)+alpha times O-delta times S-chi times S$$
(6)
To simulate a CNR intervention, an additional subpopulation was added to the system (Eq. 7): (iv) the neutered free-roaming dog population (N; neutered, free-roaming). In this simulation, a proportion of the intact (I) free-roaming dog population was removed and added to the neutered free-roaming dog population. A neutering rate (φ) was added to the differential equations describing the intact free-roaming and the neutered free-roaming dog populations. Neutering was assumed to be lifelong (e.g. gonadectomy); a neutered free-roaming dog could not re-enter the intact free-roaming dog subpopulation. Neutered free-roaming dogs were removed from the population through the density dependent neutered dog death rate (µn); death rate increased when the population was closer to the carrying capacity. The death rate was a non-linear function of population size and carrying capacity modelled using a table lookup function (Fig. S1). Neutered free-roaming dogs were also removed through adoption to the owned dog population, and we assumed that adoption rates did not vary between neutered and intact free-roaming dogs.Neutered free-roaming dog population:$$frac{dN}{dt}=varphi times I-{mu }_{n}times N-delta times N$$
(7)
Intact free-roaming dog population with neutering intervention.$$frac{dI}{dt}={r}_{s}times Itimes left(1-frac{(I+N)}{{K}_{s}}right)+alpha times O-delta times I-varphi times I$$
(8)
To simulate a responsible ownership intervention, the baseline model was applied with decreased rate of abandonment/roaming (α) and increased rate of shelter adoption (β). To simulate combined CNR and responsible ownership, a proportion of the intact free-roaming dog population was removed through the neutering rate (φ), abandonments/roaming decreased (α) and shelter adoptions increased (β). In combined CNR and sheltering interventions, a proportion of the intact free-roaming dog population (I) was removed through neutering (φ) and added to the neutered free-roaming dog population (N), and a proportion was removed through sheltering (σ) and added to the shelter dog population (H).Intact free-roaming dog population with combined CNR and sheltering interventions:$$frac{dI}{dt}={r}_{s}times Stimes left(1-frac{(I+N)}{{K}_{s}}right)+alpha times O-delta times I-varphi times I- sigma times I$$
(9)
Intervention length, periodicity, and coverageAll simulations were run for 70 years to allow populations to reach equilibrium. It is important to note that this is a theoretical model; running the simulations for 70 years allows us to compare the interventions, but does not accurately predict the size of the dog subpopulations over this long time period. Interventions were applied for two lengths of time: (i) the full 70-year duration of the simulation; and (ii) a five-year period followed by no further intervention, to simulate a single period of investment in population management. In each of these simulations, we modelled the interventions as (i) continuous (i.e. a constant rate of e.g. neutering) and (ii) annual (i.e. intervention applied once per year). Interventions were run at low, medium, and high coverages (Table 2). As the processes are not equivalent, we apply different percentages for the intervention coverage (culling/neutering/sheltering) and the percent increase/decrease in parameter rates for the responsible ownership intervention. Intervention coverage refers to the proportion of dogs that are culled/neutered/sheltered per year (i.e. 20%, 40% and 70% annually) and, for responsible ownership interventions, the decrease in abandonment/roaming rate and increase in the adoption rate of shelter dogs (30%, 60% and 90% increase/decrease from baseline values). To model a low (20%), medium (40%) and high (70%) proportion of free-roaming dogs caught, but where half of the dogs were sheltered and half were neutered-and-returned, combined CNR and sheltering interventions were simulated at half-coverage (e.g. intervention rate of 0.7 was simulated by 0.35 neutered and 0.35 sheltered). For continuous interventions, sheltering (σ), culling (χ), and CNR (φ) were applied continuously during the length of the intervention. For annual interventions, σ, χ, and φ were applied to the ordinary differential equations using a forcing function applied at 12-month intervals. In simulations that included responsible ownership interventions, the decrease in owned dog abandonment/roaming (α) and the increase in shelter adoption (β) was assumed instantaneous and continuous (i.e. rates did not change throughout the intervention).Model outputsThe primary outcome of interest was the impact of interventions on free-roaming dog population size. For interventions applied for the duration of the simulation, we calculated: (i) equilibrium population size for each population; (ii) percent decrease in free-roaming dog population; (iii) costs of intervention in terms of staff-time; and (iv) an overall welfare score. For interventions applied for a five-year period, we also calculated: (v) minimum free-roaming dog population size and percent reduction from initial population size; and (vi) the length of time between the end of the intervention and time-point at which the free-roaming dog population reached above 20,000 dogs (the assumed initial free-roaming dog population size of Lviv, based on our previous research32, see Supplementary Information for detail).The costs of population management interventions vary by country (e.g. staff salaries vary between countries) and by the method of application (e.g. method of culling, or resources provided in a shelter). To enable a comparison of the resources required for each intervention, the staff time (staff working-months) required to achieve the intervention coverage was calculated. While this does not incorporate the full costs of an intervention, as equipment (e.g. surgical equipment), advertising campaigns, travel costs for the animal care team, and facilities (e.g. clinic or shelter costs) are not included, it can be used as a proxy for intervention cost. Using data provided from VIER PFOTEN International, we estimated the average number of staff required to catch and neuter the free-roaming dog population and to house the shelter dog population in each intervention, using this data as a proxy for catching and sheltering/culling. The number of dogs that can be cared for per shelter staff varies by shelter. To account for this, we estimated two staff-to-dog ratios (low and high). Table 3 describes the staff requirements for the different interventions.Table 3 Staff required for interventions and the number of dogs processed per staff per day.Full size tableUsing the projected population sizes, the staff time required for each staff type (e.g. number of veterinarian-months of work required) was calculated for each intervention. Relative salaries for the different staff types were estimated (Table 3). The relative salaries were used to calculate the cost of the interventions by:[Staff time required × relative salary ] × €20,000.Where €20,000 was the estimated annual salary of a European veterinarian, allowing relative staff-time costs to be compared between the different interventions. Average annual costs were reported.To provide overall welfare scores for each of the interventions, we apply the estimated welfare scores on a one to five scale, for each of the dog subpopulations, as determined by Hogasen et al. (2013)22. This scale is based on the Five Freedoms (freedom from hunger and thirst; freedom from discomfort; freedom from pain, injury, or disease; freedom to express normal behaviour; freedom from fear and distress42,43) and was calculated using expert opinions from 60 veterinarians in Italy22. The scores were weighted by the participants’ self-reported knowledge of different dog subpopulations, which resulted in the following scores: 2.8 for shelter dogs (WH); 3.5 for owned dogs (WO); 3.1 for neutered free-roaming dogs (WN); and 2.3 for intact free-roaming dogs (WI)22.Using these estimated welfare scores, we calculated an average welfare score for the total dog population based on the model’s projected population sizes for each subpopulation (Eq. 10). For interventions running for the duration of the simulation, the welfare score was calculated at the time point (t) when the population reached an equilibrium size. For interventions running for five years, the welfare score was calculated at the end of the five-year intervention. The percentage change in welfare scores from the baseline simulation were reported.$$Welfare score= frac{{H}_{t}times {W}_{H}+{O}_{t}times {W}_{O}+{N}_{t}times {W}_{N}+{I}_{t}times {W}_{I}}{{H}_{t}+{O}_{t}+{N}_{t}+{I}_{t}}$$
(10)
Model validation and sensitivity analysisA global sensitivity analysis was conducted on all parameters described in the baseline simulation and all interventions applied continuously, at high coverage, for the full duration of the simulation. A Latin square design algorithm was used in package “FME”44 to sample the parameters within their range of values (Table 1). For the global sensitivity analysis on interventions, all parameter values were varied, apart from the parameters involved in the intervention (e.g. culling, neutering, abandonment/roaming rates). The effects of altering individual parameters (local sensitivity analysis) on the population equilibrium was also examined for the baseline simulation using the Latin square design algorithm to sample each parameter, individually, within their range of values. Sensitivity analyses were run for 100 simulations over 50 years solved with 0.01 step sizes. More

Lehtonen12 presents three simple models with the same broad structure: a single mutant individual with divergent mating behaviour arises in a population of ‘residents’ that all play the same strategy, and the success of that mutant is then followed (Figs. 1, 2). Specifically, Lehtonen investigates the fitness benefits of increased mating for mutant males in comparison to mutant females. Two important parameters can be varied: (i) the degree of anisogamy (defined here as the ratio of sperm number to egg number), which captures how divergent males and females are in the size (and thus number) of gametes they produce, and (ii) the efficiency of fertilisation, which determines how easily gametes can find and fuse with each other. If fertilisation is highly efficient, then gametes of the less numerous type will achieve nearly full fertilisation; on the other hand, inefficient fertilisation can result in gametes of both sexes going unfertilised.Fig. 2: Structure of the three models of Lehtonen12, showing differences in mating behaviour between resident males (green), resident females (blue) and mutant males and females (both yellow).For illustration, we suppose that females produce four eggs each and males produce eight sperm (the anisogamy ratio in nature is typically much higher). In Model 1, resident individuals spawn monogamously in a ‘nest’ (black outline), whereas mutant males and females can bring additional partners to their nest to spawn in a group. In Model 2, resident individuals divide their gametes equally among m spawning groups, each consisting of m individuals of each sex (shown here with m = 2). Mutant males and females instead divide their gametes among a larger or smaller number of groups, mmutant (shown here with mmutant = 4). In Model 3, there is a further sex asymmetry in addition to anisogamy: Fertilisation takes place inside the female’s body. Resident individuals mate with m partners (shown here with m = 2), whereas mutant males and females mate with a larger or smaller number of partners, mmutant (shown here with mmutant = 4).Full size imageIn the first two models, fertilisation is external and no assumptions are made about pre-existing differences between the sexes apart from the number of gametes they produce. In other words, males and females are identical except that males produce sperm in greater numbers than females produce eggs. In Model 1, resident individuals are assumed to mate monogamously, whereas a mutant can monopolise multiple partners of the opposite sex (Fig. 2). Importantly, both male and female mutants can bring additional partners back to their ‘nest’ to spawn in a group. When fertilisation is highly efficient, females can fertilise all of their eggs by bringing back a single male, and there is simply no benefit (in this model) of seeking further partners (Fig. 1A). In contrast, anisogamy means that males always produce at least some gametes in excess, and thus can benefit from seeking additional mates. When fertilisation is inefficient, however, both sexes benefit from increasing the concentration of opposite-sex gametes at their ‘nest’ (Fig. 1B). This latter benefit is sex-symmetric, whereas the former continues to apply only to males. As a consequence, the Bateman gradients are always steeper for males than for females (Fig. 1A, B), confirming Bateman’s argument.Model 2 similarly assumes external fertilisation, but in this case the resident males and females meet in groups consisting of m individuals of each sex (Fig. 2). Fertilisation occurs via group spawning. It is assumed that each resident individual divides its gametes evenly across M groups, whereas mutant individuals can instead spread their gametes over a larger or smaller number of groups (note that the author assumes that M = m, but this assumption could be relaxed without undermining the core argument). Spreading gametes out across a larger number of spawning groups does not increase the concentration of opposite-sex gametes they encounter (Fig. 2). However, a mutant that spreads its gametes more widely reduces the density of its own gametes across those groups in which it spawns. This in turn results in there being more opposite-sex gametes for each gamete of the mutant’s sex in those groups. For example, in Fig. 2, mutant males spawn in twice as many groups as resident males and thereby halve the density of their own sperm in each group. The resulting egg-to-sperm ratio of (frac{4}{6}=frac{2}{3}) is more favourable than the ratio of (frac{4}{8}=frac{1}{2}) that the resident males experience. Mutant females can similarly increase local sperm-to-egg ratios by spreading their eggs over more groups. However, in contrast to males, this only leads to fitness benefit if fertilisation is inefficient, and even then the benefit to females is very modest (scarcely perceptible in Fig. 1D). Gamete spreading reduces wasteful competition among the mutants’ own gametes for fertilisation. Such ‘local’ gamete competition, like gamete competition more generally, is stronger among sperm than among eggs because sperm are more numerous under anisogamy13,14. Consequently, as in Model 1, Bateman gradients are always steeper in males (Fig. 1C, D). Recall that the results of the above models emerge in the absence of any assumptions beyond the sex difference in the number of gametes produced.The third and final model allows for a further pre-existing difference between the sexes in addition to anisogamy: internal fertilisation, which is common and widespread in animals (Fig. 2)15. Each female is assumed to mate with m males, while each male divides his gametes evenly among m females. As in the previous two models, males benefit more than females from additional matings under most conditions. However, in the particular case where fertilisation is highly inefficient and the ratio of sperm to eggs is not too large, the pattern can theoretically reverse, such that female Bateman gradients exceed their male counterparts (Fig. 1F). The reason is that the effects of gamete concentration are asymmetric under internal fertilisation: Multiple mating by a female increases the local concentration of sperm its eggs experience, whereas a male’s multiple mating does not increase the concentration of eggs around its sperm (Fig. 2). Under conditions of severe sperm limitation—due to both weak anisogamy and highly inefficient fertilisation—this can lead to females benefitting more from additional matings than males (Fig. 1F). Although intriguing, it is unclear whether this finding has any empirical relevance, as sperm limitation is probably rarely severe in internal fertilisers. Under more realistic conditions of moderate to high fertilisation rates, sex differences in the degree of local gamete competition once again become decisive, and male Bateman gradients exceed their female counterparts (Fig. 1E). More

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