Now consider a generalized model in which the species interactions are heterogeneous. A natural way of introducing heterogeneity in the system is by having a species diversify into subpopulations with different interaction strengths12,13,14,15. This way of modeling heterogeneity is useful as it can describe different kinds of heterogeneity. For example, the subpopulations could represent polymorphic traits that are genetically determined or result from plastic response to heterogeneous environments. A population could also be divided into local subpopulations in different spatial patches, which can migrate between patches and may face different local predators. We can also model different behavioral modes as subpopulations that, for instance, spend more time foraging for food or hiding from predators. We study several kinds of heterogeneity after we introduce a common mathematical framework. By studying these different scenarios using variants of the model, we show that our main results are not sensitive to the details of the model.
We focus on the simple case where only the prey species splits into two types, (C_1) and (C_2), as illustrated in Fig. 1b. The situation is interesting when predator A consumes (C_1) more readily than predator B and B consumes (C_2) more readily than A (i.e., (a_1 / a_0 > b_1 / b_0) and (b_2 / b_0 > a_2 / a_0), which is equivalent to the condition that the nullclines of A and B cross, see section “Resources competition and nullcline analysis”). The arrows between (C_1) and (C_2) in Fig. 1b represent the exchange of individuals between the two subpopulations, which can happen by various mechanisms considered below. Such exchange as well as intraspecific competition between (C_1) and (C_2) result from the fact that the two prey types remain a single species.
The system is now described by an enlarged Lotka-Volterra system with four variables, A, B, (C_1), and (C_2):
$$begin{aligned} dot{A}&= varepsilon _A ,alpha _{A1} , A , C_1 + alpha _{A2} , A , C_2 – beta _A , A end{aligned}$$
(3a)
$$begin{aligned} dot{B}&= varepsilon _B , alpha _{B1} , B , C_1 + alpha _{B2} , B , C_2 – beta _B , B end{aligned}$$
(3b)
$$begin{aligned} dot{C_1}&= C_1 , (beta _C – alpha _{CC} , C)-alpha _{A1} , C_1 A-alpha _{B1} , C_1 B – sigma _1 , C_1 + sigma _2 , C_2 end{aligned}$$
(3c)
$$begin{aligned} dot{C_2}&= C_2 , (beta _C – alpha _{CC} , C) -alpha _{A2} , C_2 A -alpha _{B2} , C_2 B + sigma _1 , C_1 – sigma _2 , C_2 end{aligned}$$
(3d)
The parameters in these equations and their meanings are listed in Table 1. Here we assume that the prey types (C_1) and (C_2) have the same birth rate and intraspecific competition strength, but different interaction strengths with A and B. Note that (C_1) and (C_2) are connected by the (sigma _i) terms, which represent the exchange of individuals between these subpopulations through mechanisms studied below; these terms indicate a major difference between our model of a prey with intraspecific heterogeneity and other models of two prey species. For the convenience of analysis, we transform the variables (C_1) and (C_2) to another pair of variables C and (lambda), where (C equiv C_1 + C_2) is the total population of C as before, and (lambda equiv C_2 / (C_1 + C_2)) represents the composition of the population (Fig. 1c). After this transformation and rescaling of variables (described in “Methods”), the new dynamical system can be written as:
$$begin{aligned} dot{A}&= A , big ( C , (a_1 (1-lambda ) + a_2 lambda ) – a_0 big ) end{aligned}$$
(4a)
$$begin{aligned} dot{B}&= B , big ( C , (b_1 (1-lambda ) + b_2 lambda ) – b_0 big ) end{aligned}$$
(4b)
$$begin{aligned} dot{C}&= C , big ( 1 – C – A (a_1 (1-lambda ) + a_2 lambda ) – B (b_1 (1-lambda ) + b_2 lambda ) big ) end{aligned}$$
(4c)
$$begin{aligned} dot{lambda }&= lambda (1-lambda ) , big ( A (a_1 – a_2) + B (b_1 – b_2) big ) + eta _1 (1-lambda ) – eta _2 lambda end{aligned}$$
(4d)
Here, (a_i) and (b_i) are the (rescaled) feeding rates of the predators on the prey type (C_i); (a_0) and (b_0) are the death rates of the predators as before; (eta _1) and (eta _2) are the exchange rates of the prey types (Table 1). The latter can be functions of other variables, representing different kinds of heterogeneous interactions that we study below. Notice that Eqs. (4a–4c) are equivalent to the homogeneous Eqs. (2a–2c) but with effective interaction strengths (a_text {eff} = (1-lambda ) , a_1 + lambda , a_2) and (b_text {eff} = (1-lambda ) , b_1 + lambda , b_2) that both depend on the prey composition (lambda) (Fig. 1c).
The variable (lambda) can be considered an internal degree of freedom within the C population. In all of the models we study below, (lambda) dynamically stabilizes to a special value (lambda ^*) (a bifurcation point), as shown in Fig. 3a. Accordingly, a new equilibrium point (P_N) appears (on the line (mathscr {L}) in Fig. 2), at which all three species coexist. For comparison, Fig. 3b shows the equilibrium points if (lambda) is held fixed at any other values, which all result in the exclusion of one of the predators. Thus, heterogeneous interactions give rise to a new coexistence phase (see Fig. 4 below) by bringing the prey composition (lambda) to the value (lambda ^*), instead of having to fine-tune the interaction strengths. The exact conditions for this new equilibrium to be stable are detailed in “Methods”.
Inherent heterogeneity
We first consider a scenario where individuals of the prey species are born as one of two types with a fixed ratio, such that a fraction (rho) of the newborns are (C_2) and ((1-rho )) are (C_1). This could describe dimorphic traits, such as the winged and wingless morphs in aphids12 or the horned and hornless morphs in beetles13. We call this “inherent” heterogeneity, because individuals are born with a certain type and cannot change in later stages of life. The prey type given at birth determines the individual’s interaction strength with the predators. This kind of heterogeneity can be described by Eq. (4d) with (eta _1 = rho (1-C)) and (eta _2 = (1-rho ) (1-C)) (see “Methods”).
The stable equilibrium of the system can be represented by phase diagrams that show the identities of the species at equilibrium. We plot these phase diagrams by varying the parameters (a_2) and (b_1) while keeping (a_1) and (b_2) constant. As shown in Fig. 4a–d, the equilibrium state depends on the parameter (rho). In the limit (rho = 0) or 1, we recover the homogeneous case because only one type of C is produced. The corresponding phase diagrams (Fig. 4a, d) contain only two phases where either of the predators is excluded, illustrating the competitive exclusion principle. For intermediate values of (rho), however, there is a new phase of coexistence that separates the two exclusion phases (Fig. 4b, c). There are two such regions of coexistence, which touch at a middle point and open toward the bottom left and upper right, respectively. The middle point is at ((a_2/a_0 = b_2/b_0, b_1/b_0 = a_1/a_0)), where the feeding preferences of the two predators are identical (hence their niches fully overlap). Towards the origin and the far upper right, the predators consume one type of C each (hence their niches separate). The coexistence region in the bottom left is where the feeding rates of the predators are the lowest overall. There can be a region (yellow) where both predators go extinct, if their primary prey type alone is not enough to sustain each predator. Increasing the productivity of the system by increasing the birth rate ((beta _C)) of the prey eliminates this extinction region, whereas lowering productivity causes the extinction region to take over the lower coexistence region. Because the existence and identity of the phases is determined by the configuration of the equilibrium points (Fig. 2, see also section “Mathematical methods”), the qualitative shape of the phase diagram is not sensitive to changes of parameter values.
The new equilibrium is not only where the predators A and B can coexist, but also where the prey species C grows to a larger density than what is possible for a homogeneous population. This is illustrated in Fig. 3b, which shows the equilibrium population of C if we hold (lambda) fixed at different values. The point (lambda = lambda ^*) is where the system with a dynamic (lambda) is stable, and also where the population of C is maximized (when A and B prefer different prey types). That means the population automatically stabilizes at the optimal composition of prey types. Moreover, the value of (C^*) at this coexistence point can even be larger than the equilibrium population of C when there is only one predator A or B. This is discussed further in section “Multiple-predator effects and emergent promotion of prey”. These results suggest that heterogeneity in interaction strengths can potentially be a strategy for the prey population to leverage the effects of multiple predators against each other to improve survival.
Reversible heterogeneity
We next consider a scenario where individual prey can switch their types. This kind of heterogeneity can model reversible changes of phenotypes, i.e., trait changes that affect the prey’s interaction with predators but are not permanent. For example, changes in coat color or camouflage14,16,17, physiological changes such as defense15, and biomass allocation among tissues18,19. One could also think of the prey types as subpopulations within different spatial patches, if each predator hunts at a preferred patch and the prey migrate between the patches20,21. With some generalization, one could even consider heterogeneity in resources, such as nutrients located in different places, that can be reached by primary consumers, such as swimming phytoplankton22. We can model this “reversible” kind of heterogeneity by introducing switching rates from one prey type to the other. In Eq. (4d), (eta _1) and (eta _2) now represent the switching rates per capita from (C_1) to (C_2) and from (C_2) to (C_1), respectively. Here we study the simplest case where both rates are fixed.
In the absence of the predators, the natural composition of the prey species given by the switching rates would be (rho equiv eta _1 / (eta _1 + eta _2)), and the rate at which (lambda) relaxes to this natural composition is (gamma equiv eta _1 + eta _2). Compared to the previous scenario where we had only one parameter (rho), here we have an additional parameter (gamma) that modifies the behavior of the system. Fig. 4e–h shows phase diagrams for the system as (rho) is fixed and (gamma) varies. We again find the new equilibrium (P_N) where all three species coexist. When (gamma) is small, the system has a large region of coexistence. As (gamma) is increased, this region is squeezed into a border between the two regions of exclusion, where the slope of the border is given by (eta _1/eta _2) as determined by the parameter (rho). However, this is different from the exclusion we see in the case of inherent heterogeneity, which happens only for (rho rightarrow 0) or 1, where the borders are horizontal or vertical (Fig. 4a,d). Here the predators exclude each other despite having a mixture of prey types in the population.
This special limit can be understood as follows. For a large (gamma), (lambda) is effectively set to a constant value equal to (rho), because it has a very fast relaxation rate. In other words, the prey types exchange so often that the population always maintains a constant composition. In this limit, the system behaves as if it were a homogeneous system with effective interaction strengths (a_text {eff} = (1-rho ) , a_1 + rho , a_2) and (b_text {eff} = (1-rho ) , b_1 + rho , b_2). As in a homogeneous system, there is competitive exclusion between the predators instead of coexistence. This demonstrates that having a constant level of heterogeneity is not sufficient to cause coexistence. The overall composition of the population must be able to change dynamically as a result of population growth and consumption by predators.
An interesting behavior is seen when we examine a point inside the shrinking coexistence region as (gamma) is increased. Typical trajectories of the system for such parameter values are shown in Fig. 5. As (gamma) increases, the system relaxes to the line (mathscr {L}) quickly, then slowly crawls along it towards the final equilibrium point (P_N). This is because increasing (gamma) increases the speed that (lambda) relaxes to (lambda ^*), and when (lambda rightarrow lambda ^*), (mathscr {L}) becomes marginally stable. Therefore, the attraction to (mathscr {L}) in the perpendicular direction is strong, but the attraction towards the equilibrium point along (mathscr {L}) is weak. This leads to a long transient behavior that makes the system appear to reach no equilibrium in a limited time23,24. It is especially true when there is noise in the dynamics, which causes the system to diffuse along (mathscr {L}) with only a weak drift towards the final equilibrium (Fig. 5). Thus, the introduction of a fast timescale (quick relaxation of (lambda) due to a large (gamma)) actually results in a long transient.
Adaptive heterogeneity
A third kind of heterogeneity we consider is the change of interactions in time. By this we mean an individual can actively change its interaction strength with others in response to certain conditions. This kind of response is often invoked in models of adaptive foraging behavior, where individuals choose appropriate actions to maximize some form of fitness25,26. For example, we may consider two behaviors, resting and foraging, as our prey types. Different predators may prefer to strike when the prey is doing different things. In response, the prey may choose to do one thing or the other depending on the current abundances of different predators. Such behavioral modulation is seen, for example, in systems of predatory spiders and grasshoppers27. Phenotypic plasticity is also seen in plant tissues in response to consumers28,29,30.
This kind of “adaptive” heterogeneity can be modeled by having switching rates (eta _1) and (eta _2) that are time-dependent. Let us assume that the prey species tries to maximize its population growth rate by switching to the more favorable type. From Eq. (4c), we see that the growth rate of C depends linearly on the composition (lambda) with a coefficient (u(A,B) equiv (a_1 – a_2) A + (b_1 – b_2) B). Therefore, when this coefficient is positive, it is favorable for C to increase (lambda) by switching to type (C_2). This can be achieved by having a positive switching rate (eta _2) whenever (u(A,B) > 0). Similarly, whenever (u(A,B) < 0), it is favorable for C to switch to type (C_1) by having a positive (eta _1). In this way, the heterogeneity of the prey population constantly adapts to the predator densities. We model such adaptive switching by making (eta _1) and (eta _2) functions of the coefficient u(A, B), e.g., (eta _1(u) = 1/(1+mathrm {e}^{kappa u})) and (eta _2(u) = 1/(1+mathrm {e}^{-kappa u})). The sigmoidal form of the functions means that the switching rate in the favorable direction for C is turned on quickly, while the other direction is turned off. The parameter (kappa) controls the sharpness of this transition.
Phase diagrams for the system with different values of (kappa) are shown in Fig. 4i–l. A larger (kappa) means the prey adapts its composition faster and more optimally, which causes the coexistence region to expand. In the extreme limit, the system changes its dynamics instantaneously whenever it crosses the boundary where (u(A,B) = 0), like in a hybrid system31. Such a system can still reach a stable equilibrium that lies on the boundary, if the flow on each side of the boundary points towards the other side32. This is what happens in our system and, interestingly, the equilibrium is the same three-species coexistence point (P_N) as in the previous scenarios. The region of coexistence turns out to be largest in this limit (Fig. 4l).
Our results suggest that the coexistence of the predators can be viewed as a by-product of the prey’s strategy to maximize its own benefit. The time-dependent case studied here represents a strategy that involves the prey evaluating the risk posed by different predators. This is in contrast to the scenarios studied above, where the prey population passively creates phenotypic heterogeneity regardless of the presence of the predators. These two types of behavior are analogous to the two strategies studied for adaptation in varying environments, i.e., sensing and bet-hedging33,34. The former requires accessing information about the current environment to make optimal decisions, whereas the latter relies on maintaining a diverse population to reduce detrimental effects caused by environmental changes. Here the varying abundances of the predators play a similar role as the varying environment. From this point of view, the heterogeneous interactions studied here can be a strategy of the prey species that is evolutionarily favorable.
Source: Ecology - nature.com