A continuous ideal free distribution approach to the dynamics of selfish, cooperative and kleptoparasitic populations

Population distributions depend upon the aggregate behavioural responses of individuals to a range of environmental factors. We extend a model of ideally motivated populations to describe the local and regional consequences of interactions between three populations distinguished by their levels of cooperation and exploitation. Inspired by the classic prisoner's dilemma game, stereotypical fitness functions describe a baseline non-cooperative population whose per capita fitness decreases with density, obligate co-operators who initially benefit from the presence of conspecifics, and kleptoparasites who require heterospecifics to extract resources from the environment. We examine these populations in multiple combinations, determine where both local and regional coexistence is permitted, and investigate conditions under which one population will invade another. When they invade co-operators in resource-rich areas, kleptoparasites initiate a dynamic instability that leads to the loss of both populations; however, selfish hosts, who can persist at low densities, are immune to this risk. Furthermore, adaptive movement may delay the onset of instability as dispersal relieves dynamic stress. Selfish and cooperative populations default to mutual exclusion, but asymmetric variations in interference strength may relax this condition and permit limited sympatry within the environment. Distinct sub-communities characterize the overall spatial structure.

1. Introduction

Competition, cooperation and other interactions influence the persistence or exclusion of populations within a community. Individual responses to these factors—and others such as resource availability, environmental quality and risk of predation—shape distributions over regional scales [1]. Competition's role in setting ranges is increasingly understood [2–6], but our theoretical knowledge of the role that cooperation, parasitism and intra-guild predation play is just beginning (e.g. [7]). Parasitism, for example, affects nearly all species, and empirically it influences the structuring of communities [8–10], yet corresponding mathematical studies remain sparse. In counterpoint, cooperation and its vulnerability to exploitation by cheaters has inspired considerable work by game theorists using various extensions of the classic prisoner's dilemma (e.g. [11–13]), but the resulting models often lack a biological foundation. In this paper, we employ a modelling framework that bridges game theory and ecological modelling to study the pairwise and collective interactions of three populations characterized by distinct levels of cooperation or parasitism. We examine both the local population dynamics and the spatial arrangements that arise when individuals adaptively disperse across the landscape.

Consumers of common resources primarily interact via competition through territoriality, interference and resource depletion. The frequency of competitive encounters varies from solitary animals that maintain relatively large territorial ranges [14–16] to groups of more social species which may conflict while patrolling borders [17]. Close proximity within-group can also be a source of interference and produce physical barriers to movement, increase the incidence of conspecific attacks or degrade habitat quality. Illustrating that last possibility, roe deer (Capreolus capreolus) trample patches during highly concentrated foraging during winter [18]. Even when there are secondary benefits for associating (herd protection, predator alarms, etc.), individuals largely survive by their own efforts.

Other social species do demonstrate cooperation in their actions. Eusociality in insects is the most recognized example of animal cooperation in which individuals sacrifice their own reproductive potential to aid conspecifics or raise the offspring of others [19]. Red harvester ants, Pogonomyrex barbatus, employ an age-dependent division of labour with young workers engaged in brood care and older workers foraging outside of the nest [20]. Individual survival is often low for obligate co-operators [21], and a minimum group size is required for successful activity. Common predatory pack animals improve hunting efficiency and can bring down larger prey by increasing pack size, while African wild dogs, Lycaon pictus, are also obligate cooperative breeders that need conspecific helpers for foraging, breeding, and deterring natural enemies [22]. Dolphins and killer whales are known to herd fish into bait balls through carouseling [23], while humpbacks utilize bubble netting to similar effect with krill [24]. Such activities also allow other predators (e.g. swordfish, sharks and gulls) to rush in and claim a share of the prey. Cooperative benefits are not limitless, however, and continued increases to group size or labour division may negatively affect the overall health of the population as individuals' shares of resources decline.

A number of possible mechanisms supporting the promotion of cooperation have emerged from game theoretic studies. These include kin selection [25], partner selection [26,11], role-model selection [12], group size, age structure and memory [27–29], social diversity and variation of the strength of cooperation [30], and tit-for-tat strategies [31]. See Wang et al. [13] and citations therein for an extensive review of game theoretic treatments. Experimental results also point to spatial structure and expansion waves as mechanisms maintaining cooperation [32].

Contra cooperation, kleptoparasitism occurs when one animal surreptitiously or aggressively steals resources (e.g. food, nesting material) gathered by another rather than obtaining them independently [33]. Kleptoparasitism is a common phenomenon throughout the taxa [8,34,35]. Examples can be found among birds (frigate-birds, Fregata spp., and skuas, Stercorarius spp. [36]), hymenopterans like Spinola bees, Radoszkowskiana rufiventris [37], and the wasp Argochrysis armilla [38], spiders like Argyrodes elevatus [39], fish (the Western Buffalo bream, Kyphosus cornelii [40]) and mammals (the spotted hyena, Crocuta crocuta) [41], and even budding yeast where one strain can catalyse the conversion of sucrose and another cannot [32]. Some kleptoparasites force their host-victims to share items or drive them away completely ([42], and citations therein). Hyenas and lions claim kills taken by others who cannot successfully defend their spoils against interlopers. Similarly, Arctic skuas intercept auks returning to their nests with fish [33], and the gecko Phelsuma inexpectata steals bee pollen [43]. Less confrontational, fork-tailed drongos use false alarm calls to parasitize both birds and small mammals ([44]; modelled by [45]), a tactic also used by shrike tanagers in aerial tumbles [46], while red-faced spinetails steal untended nest material [35]. Even brood parasitism is a variation on this general concept, notable in the Apidae bee family [9,47] and cowbirds. The shiny cowbird (Molothrus bonariensis) has endangered the yellow-shouldered blackbird (Agelaius xanthomus) in Puerto Rico since 1976 [48], and in general brood parasitism threatens several species of birds [49].

In this paper, we extend a reaction–advection model of ideally motivated populations [2,3] to describe interactions between three stereotypical populations consisting of selfish individuals, co-operators and kleptoparasites. We consider these populations in isolation and pairwise interactions before concluding with an examination of the dynamics when all three are present. Through a combination of analytic and simulation approaches, we detail the conditions under which one group can invade another and where persistent local and regional coexistence is possible. Kleptoparasites may invade any host population of sufficient size, which is functionally invariant across host types. Where invasion is possible, kleptoparasites and selfish hosts always reach stable coexistence; however, stability becomes resource-dependent with cooperative hosts. High resources excite oscillations in population levels that eventually depress host density below a sustainability threshold, culminating in the loss of both populations; however, adaptive movement may delay the onset of instabilities. Selfish and cooperative populations are intrinsically mutually exclusive at the local level. Asymmetries in interference strengths can relax this situation and permit limited local sympatry (cf. [3]), but increasing resources will reestablish exclusion. Finally, regional communities are spatially structured, with local compositions stratified by resource values.

2. Model

This section extends a reaction–dispersal model for ideally motivated populations [2] to three interacting populations distinguished by varying degrees of cooperation and exploitation. Ideally motivated individuals migrate in the direction of greatest immediate increase in fitness in a continuous analogue to the ideal free distribution ([50]; also see [51]). Only some portions of total fitness are directly measurable by individuals, e.g. resource availability and competition levels, while others, such as mortality risks, may be obscured. Imperfect knowledge of the spatial variation in those unobservable factors leads to source-sink dynamics ([2]; J.T.R. 2016, unpublished manuscript); however, this paper will not explore that aspect of the model.

Consider a population with local density u_i(x,t) at position x and time t that is subject to an observable fitness function f_i(R,u_i,*). This function represents the per capita amount of resources extracted from the environment based upon local resource availability R(x), conspecific density u_i, plus other factors including heterospecifics. Gathered resources are converted to reproductive growth with metabolic efficiency r_i, and the population suffers a uniform per capita mortality rate μ_i. These last two processes are non-observable and do not affect movement. Density flows in the direction of improved fitness and results in a net change due to immigration equal to −k_i∇⋅(u_i∇f_i), where k_i is the sensitivity to local changes in the fitness landscape. The complete equation of change for population density is thus

2.1

In a local, non-spatial model, the corresponding system of ordinary differential equations (ODEs) would be

2.2

2.1 Dynamic and spatial equilibrium

Per Rowell [2], a rapid or highly motile population described by model equation (2.1) approaches a distribution wherein fitness has a uniform value f_i=E_i wherever u_i>0 and f_i≤E_i at locations immediately neighbouring the inhabited region (figure 1a,b). Even when multiple species are admitted regionally, the populations generate jointly ideal distributions [3]. Local dynamic equilibrium is achieved either when u_i=0 or f_i=μ_i/r_i. The latter is also consistent with balanced migration for ideally motivated competitors.

2.2 Selfishness, cooperation and exploitation

The communities under consideration are composed of two or three populations whose fitness functions are stereotypical expressions of selfishness, cooperation or exploitation (i=1,2 and 3, respectively),

2.3a

2.3b

2.3c

Rowell [2] previously described the first two host populations in isolation (u_j=0,j≠i). Direct competition between the two hosts is a novel development, as is the introduction of the exploitative functional type to this model framework (i=3). The host functions derive from first and second degree saturation curves describing total grazing efforts with interference interpreted as prolonging resource retrieval time. h₁ and correspond to intrinsic resource collection times. In this model, a scramble kleptoparasitism transfers a portion θ_i of gathered resources upon encounters with host-type individuals (e.g. via false alarm calls, combative threat or surreptitious theft).

The population of ‘selfish’ or ‘self-reliant’ individuals (i=1) has been studied with regard to range limits [2], performance trade-offs [3] and harvesting (J.T.R. 2016, unpublished manuscript). Increased density reduces individual fitness by prolonging the time required to obtain local resources due to interference competition (equation (2.3a)), ∂f₁/∂u₁<0. These populations are excellent pioneer species and disperse across heterogeneous landscapes to every contiguous location where resources sustain a local population, R>μ₁h₁/r₁. Previous theoretical results, heuristic arguments and numerical simulations strongly supported the conclusion that the uniformly-fit distribution

2.4

is the global attractor for the population under (2.1). Here, a₁₁ is the selfish intraspecific interference strength and E*₁=μ₁/r₁ is the ratio of mortality to metabolic efficiency.

By contrast, ‘contributors’ (u₂) co-operate to increase personal fitness by more effectively gathering resources, hunting larger prey or sharing nest-site responsibilities. The corresponding fitness function features an Allee effect (2.3b), and initially increases with density before declining as resources become exhausted. This function generates a strong tendency towards aggregation into clusters, which can result in spatial instabilities and a potential absence of well-posedness in the model similar to chemotaxis. Unlike that phenomenon, however, self-regulation in the present model prohibits unbounded growth. This instability is ecologically meaningful as it indicates mosaics or patchy spatial distribution across the landscape. For any uniform fitness value E₂ (with the dynamic equilibrium value E*₂=μ₂/r₂), equation (2.1) admits two principal ideal distributions (figure 1b)

2.5

In practice, a movement-balanced distribution can be patchy and draw from both ideal curves and the trivial solution. The unstable lower positive solution for (denoted ) constitutes a minimum threshold for local persistence in the ODE problem (2.2). This threshold decreases as resources improve, but because of reliance upon conspecifics, contributors are poor colonizers both locally and regionally.

The final population type (u₃) represents an obligate kleptoparasite that steals resources gathered by either of the previous two populations. This kleptoparasite cannot independently retrieve resources from the environment and is entirely dependent upon the presence of heterospecifics for survival. A fraction of resources () are transferred upon each encounter with heterospecifics, which occurs under simple mass action encounter rates (u₃u_j). The parasitized host's fitness (2.3a,b) is commensurately reduced by a factor of (1−θ_ju₃).

3. Results

3.1 Kleptoparasitic invasions

Kleptoparasites persist only in the presence of heterospecifics engaged in resource recovery; therefore, they are placed in the role of invader in any pairwise interaction (summarized in table 1). We assume that a host population has already established its intrinsic ideal equilibrium, . For cooperative hosts, which exhibit two solutions for any given fitness level, the locally attracting upper solution will be chosen. Parasite density depresses host fitness through the transfer of resources (∂f_i/∂u₃<0); however, conspecific density has no direct effect on the parasites' own fitness (∂f₃/∂u₃=0). Self-regulation is an indirect consequence of the reduction in host density.

Two invasion features are consistent across hosts. First, there is a minimum host density required to promote an initial incursion. Host self-regulation also downregulates parasites. When parasite density is initially small (u₃≈0), the host fitness equals E*_i=μ_i/r_i, and the parasite invades if the isolated host density exceeds a threshold set by the relative equilibrium fitness levels (μ_i/r_i) and the transfer rate θ_i,

3.1

Second, when coexistence occurs at the non-trivial equilibrium , the parasite equilibrium density is defined relative to host density by a simple linear equation

3.2

The appendix A details the derivation of these general results.

If the host population consists of selfish individuals (i=1), the minimum density condition (3.1) may be equivalently restated as a threshold condition on the local resource levels

3.3

which surpasses that required to sustain the host, R_via=h₁μ₁/r₁. These resource levels delimit regions of the environment supporting one of three possible outcomes for local dynamics. In areas of extremely low resources (R≤R_via) neither population is present. At intermediate resource levels (R_via<R≤R_inv), the selfish host persists alone as its density is insufficient to support kleptoparasitism (figure 2a). In regions of abundant resources (R>R_inv), parasitized coexistence is sustainable (figure 2b). When adaptive movement is incorporated into the model, the colonizing capacity of selfish individuals ensures that the hosts retain their full natural range, with the area of coexistence centralized about resource peaks and a host refuge in resource-limited areas. At the jointly ideal equilibrium distribution, resource peaks coincide with peaks in parasite density, while the host distribution exhibits a conspicuous depression in these areas (figure 1c). Numerical simulations (see appendix Ae), such as those shown in figure 3a–d and 2a,b, and the absence of additional local attractors from which to construct a spatial distribution strongly support ideal regional coexistence as a global attractor. The relative speed of movement shapes the evolution of the community. In relatively motile communities (k≥r, μ), invading kleptoparasite centralizes before expanding and displacing the host (figure 3c,d). If the community is more reactive (r, μ>k), the invasion resembles a travelling wave (figure 3a,b). Moreover, the adaptive movement dampens or eliminates the spiralling observed in the non-spatial local dynamics.

The invasion condition (3.1) remains identical when the host population is composed of co-operators; however, the region vulnerable to invasion is set by different resource levels (table 1), which may or may not exceed that required for a viable host population. The most marginalized co-operator population boasts a density , a level which is remarkably independent of the metabolic efficiency and mortality of the host. The loss of a refuge for co-operators occurs when the theft parameter exceeds

3.4

Typically, the resource value at which parasitic invasions occur exceeds the host-viability level; however, when (3.4) holds, the lower-resource refuges do not exist, and the local host population is everywhere at risk of non-recoverable destabilization (see below).

The most important difference between hosts is that host–parasite coexistence is locally stable only under intermediate resources for cooperative hosts. At particularly high resources, R>2(a₂₂/θ₂)(μ₃/r₃), local coexistence destabilizes, and the population trajectory spirals out until it crosses the minimum population threshold necessary for co-operators to persist (figure 2d), after which both the host and parasite irrevocably decline and are lost. The transient phase characterized by the expanding population spirals can be of considerable duration.

In the spatial model, the destabilization at high resources and the population's natural inclination toward clustering render large swathes of in principle viable habitat vacant in an ideal community (figure 1d). The unstable local dynamics may couple with adaptive movement (and its potential for spatial instability) to produce patchy distributions that only broadly match the ideal distribution (figure 3e–h). In more motile communities, the initiation of excitement is delayed, as movement relieves dynamic pressures and maintains a near-ideal community distribution until central hosts are pushed towards the persistence threshold.

3.2 Host exclusion

Local competition between selfish and cooperative populations leads to one of four distinct dynamic scenarios. Figure 4 provides representative phase portraits for each case. The non-trivial nullcline of each population is parabolic, one symmetric about u₂=0 (selfish) and the other symmetric about u₂=Rr₂/(2a₂₂μ₂) (cooperative). The primary outcome features mutual exclusion of the hosts with coexistence only at a saddle. This results when interference strengths are balanced (a₁₁a₂₂=a₁₂a₂₁) and the co-operators' upper solution is resistant to invasion. The three remaining cases are strict dominance by self-reliant individuals, mutual exclusion where the saddle occurs at the minimally viable semi-trivial solution , and a dynamic that admits two distinct coexistence equilibria. In this last scenario, asymmetric exclusion favours the selfish population, which still cannot be invaded, while the second coexistence state is a local attractor when co-operators are more established. In general, selfish hosts may invade established co-operators when

3.5

These last three cases are tenuously modulated by variations in intra- and inter-specific interference which influence the width and/or height of the nullclines. As resource levels increase, the dynamics always transition to the mutual exclusion of the two host populations. This development rests largely on changes to the minimum viable co-operator population—which goes towards 0—and the maximum co-operator population, whose growth outpaces the expansion of the selfish population's nullcline (R versus ). Because of this dependence on resources and interference strengths, different locations may exhibit distinct dynamic behaviours, with local dominance, coexistence and mutual exclusion all present. Figure 3i,j illustrates the community's spatial structure under a selfish invasion of established co-operators with weak inter-specific interference. Although marginalized, selfish invaders establish their own exclusive territory and co-opt or share neighbouring co-operators' territory. The resident hosts maintain a firewall at higher resources, so the selfish group cannot populate the far end of the environment.

3.3 Community dynamics

In a local, non-spatial model of community dynamics, kleptoparasitism introduces oscillatory or spiralling behaviour in the evolution of host–parasite levels. Under normalized interference (a_ij=1), host exclusion is a characteristic but not persistent feature of the local dynamics. The presence of kleptoparasites depresses the densities of both hosts by creating an apparent reduction in resource values. This drop in host density may be sufficient to guarantee selfish dominance or at least selfish promotion. For low levels of kleptoparasitism, the hosts maintain their mutual exclusion and there are two locally stable semi-trivial solutions featuring the parasite and one of the host types (figure 5a). The Allee effect ensures remains locally stable, but should parasitism result in the condition

3.6

then we arrive at an interesting scenario where the cooperative host–parasite equilibrium can remain locally stable under the restriction u₁=0, but it is unstable within the community dynamics as selfish hosts can invade (figure 5b). There is no corresponding analogous behaviour with the semitrivial solution because of the Allee effect imposed on co-operators.

Cooperative clustering limits commentary about the spatial model's global asymptotic behaviour, but we can broadly speak of regions where general phenomena are likely to be encountered. For illustrative purposes (figure 6), consider a scenario of sequential introductions: first co-operators, then parasites and finally selfish individuals. The community reaches an ideal steady state distribution before each new population is introduced at low densities. Upon the arrival of selfish individuals, the community resembles figure 1d, and they perceive two relatively high-valued areas ( f₁) that could attract colonization. First, selfish individuals can monopolize untapped marginal resources at the periphery of the co-operators' range. The second potential aggregation site centres around the resource peak, which may or may not be vacated as a consequence of the preceding kleptoparasitic invasion. The co-operators' parasite-free zone isolates these areas during migration. If selfish indviduals arrive at the outer margins, they persist there with little change to the rest of the community (not pictured). Should they instead appear within the parasitized sub-population (e.g. a random transplant or mutation event), the depression in co-operator density provides a clear migratory path to the vacant centre (figure 6b). The parasite responds to this new prey slowly (figure 6c). The selfish expansion wave also disrupts resident co-operators in the distant sub-group upon contact, but they are not lost (figure 6d). The community nears an ideal joint distribution at the end of the simulation, (figure 6a).

In general, the community will evolve towards a constellation of distinct sub-communities. In order of increasing resources, possible areas are described as uninhabited, selfish-only, co-operator-only, parasitized co-operators and finally parasitized selfish individuals. The existence and precise width of each of these respective zones depends upon the various model parameters as well as the initial distributions of populations and the potential spatial instabilities of motivated co-operators. As in the previous section, variation of the interference strength a_ij further opens the possibility of host coexistence. The dynamic behaviour of the two host populations remains resource-mediated, so alternative spatial distributions, potentially including coexistence of hosts or all three populations, can be generated in different areas of the environment (not pictured).

4. Discussion

This paper extends a model of ideally motivated populations [2] to niche or guild competitors with fundamental functional differences in the fitness of their respective members. We elected to use three distinct fitness functions that were ecologically meaningful and captured features of the prisoner's dilemma from game theory. Per the name, co-operators benefited from the presence of conspecifics through an Allee effect, while selfish individuals and kleptoparasites were alternate versions of the ‘Defect’ strategy. At low densities, co-operators are ineffective in both resource recovery and deterrence of niche competitors, and the relationship to selfish rivals was more akin to an ammensal threat. Kleptoparasites represented a clear case of exploitation. These obligate parasites could not extract native resources on their own and negatively affected host fitness, but conspecifics imposed no direct reduction on the parasites' own fitness. Fitness loss came indirectly as host levels were depressed (see grazing model in [2]). A spatial structure for the regional community naturally emerged from the combination of adaptive movement and local dynamics.