Preference and switching in the Kill-the-Winner functional response: Diversity, size structure, and synergistic grazing in plankton models

Grazing by zooplankton can maintain diversity in phytoplankton communities by allowing coexistence between competitors in situations that would otherwise lead to competitive exclusion. In mathematical models, grazing is represented by a functional response that describes the consumption rate by an individual zooplankter as a function of phytoplankton concentration. Since its initial description, the Kill-the-Winner functional response has been increasingly adopted for large-scale biogeochemical modeling. Here, we analyze how two properties of the Kill-the-Winner functional response—preference and switching—interact to promote coexistence and increase diversity in two simple models: a diamond-shaped nutrient-phytoplankton-zooplankton model and a size-structured phytoplankton community model. We found that, compared to preference, switching leads to coexistence and increased diversity over a much wider range of environmental conditions (nutrient supply and mixing rate). In the absence of switching, preference only allows for coexistence within the narrow range of environmental conditions where the preference is precisely balanced against the competitive difference between phytoplankton types. We also explored a counterintuitive aspect of the Kill-the-Winner functional response that we have termed “synergistic grazing”. Synergistic grazing occurs when the grazing rate on one phytoplankton type increases as the biomass of an alternative phytoplankton type increases. This unrealistic effect is most evident when switching is strong and when zooplankton have a preference for the weaker competitor.

Phytoplankton communities commonly consist of many species living together in an 2 apparently homogeneous environment and competing for a small number of limiting 3 resources. This coexistence perplexed ecologists in the middle of the twentieth century 4 because the current understanding of ecology led them to believe that the strongest 5 competitor should exclude all others [1]. Hutchinson [2] termed this disparity between 6 observations and ecological theory the "paradox of the plankton." Beginning in the attack rate and handling time of the grazer for a given resource. However, functional 22 responses that include multiple resources are more complicated because the relationship 23 between consumption rate and resource density may be different for different resources. 24 These relationships may be further complicated by the fact that the consumption rate 25 of one resource may be affected by the abundance of other available resources [14,15]. 26 Gentleman et al. [16] provide a review of functional responses that have been used in 27 the literature to describe zooplankton grazing on multiple phytoplankton types. 28 The relationship between zooplankton grazers and different phytoplankton types in 29 the environment can be complex. When considering different mechanisms by which 30 zooplankton may promote coexistence between competing phytoplankton, it is helpful 31 to be precise about terminology for specific grazer characteristics. A zooplankter is said 32 to exhibit preference for a phytoplankton type when the proportion of that type in its 33 diet exceeds the proportion of that type in the environment [12,17,18]. A zooplankter is 34 said to exhibit switching when the proportion of a phytoplankton type in its diet 35 changes from less than expected to greater than expected as the proportion of that 36 phytoplankton type in the environment increases [18]. Functional responses that 37 incorporate preference and switching have been included in models to test whether 38 these behaviors promote coexistence between competing phytoplankton species or size 39 classes in circumstances that would otherwise lead to competitive exclusion [16,19]. 40 Global biogeochemical models with these kinds of functional responses show increased 41 phytoplankton diversity compared to those that do not include these behaviors [8,20]. 42 Preference and switching are commonly included simultaneously in these models, 43 however, and the individual impacts of each characteristic are rarely considered. 44 Functional responses that describe zooplankton grazing on multiple phytoplankton 45 types are usually phenomenological and their parameters often cannot be measured 46 directly. Some of these functional responses have characteristics that are biologically 47 unrealistic [16]. One such characteristic is antagonistic feeding, in which a zooplankter 48 has a higher total consumption rate when feeding on a single phytoplankton type than 49 it would when the same total phytoplankton biomass is divided among many different 50 types [21]. To avoid antagonistic feeding, Vallina et al. [20] developed an alternative 51 formulation, the Kill-the-Winner (KTW) functional response. Since its publication, the 52 KTW functional response has become a popular way to include switching behavior in a 53 variety of models [22][23][24][25][26][27][28][29][30][31][32]. 54 Vallina et al. [20] used one case of the KTW functional response to study how grazer 55 switching promotes diversity on a global scale in a size-structured phytoplankton 56 community model. Their model included the transport of plankton via a global 57 circulation model. Here, we eliminate spatial dispersal in order to isolate and focus on 58 biological interactions on the scale of competition between individual phytoplankton 59 types. We have broken down our analysis to consider the capacity of zooplankton 60 preference and switching as independent mechanisms for mediating coexistence between 61 competing phytoplankton. We also identify a specific characteristic of the KTW 62 functional response, which we refer to as "synergistic grazing", that can generate Whenever multiple phytoplankton types are available for consumption, a zooplankter's 67 consumption rate on a particular type may depend on its preference for that type.

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Zooplankton preference may arise from differential searching rates or rejection of less 69 desirable phytoplankton types [17]. By definition, preference is fixed and independent of 70 the distribution of phytoplankton types in the environment [18]. A zooplankter's 71 consumption rate on a given type may also depend upon the abundance of other 72 available types in the environment via switching. Switching represents some behavioral 73 change in the zooplankton that occurs in response to a variable phytoplankton 74 community. Such responses include changing feeding strategies or learning how to 75 capture or handle a particular phytoplankton type more efficiently [18,20].

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The KTW functional response includes both preference and switching. It defines the 77 per capita grazing rate on phytoplankton type P i as [20]. G i has two components. One component, g max n j=1 ρ j P j /(k sat + n j=1 ρ j P j ), 79 represents the total grazing rate as it depends upon the total preference-weighted  work, the KTW functional response has only been examined for α = 2 [20].

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If the zooplankter exhibits neither preference nor any switching behavior, then the 94 proportion of a phytoplankton type in the zooplankter's diet will be equal to that type's 95 proportion in the environment (Fig 1, solid  preference for a phytoplankton type, but no switching, then the proportion of a 97 phytoplankton type in the zooplankter's diet will be higher than its proportion in the proportion of a phytoplankton type in the zooplankter's diet will change from less than 100 expected to greater than expected (relative to the case where the zooplankter has no 101 preference or switching) as the proportion of that type in the environment increases 102 (Fig 1, dashed lines). Switching to the more common phytoplankton type occurs at a 103 lower proportional abundance if the zooplankter has a preference for that type, and at a 104 higher proportional abundance if it has a preference for other types. Characteristics of the KTW functional response using different combinations of preference and switching. The proportion of P 1 in a zooplankter's diet as a function of the proportion of P 1 in the environment based on Eq 1 and assuming two phytoplankton types. The different curves show different combinations of preference and switching: (a) neither preference nor switching (ρ 1 = ρ 2 , α = 1), (b) switching without preference (ρ 1 = ρ 2 , α > 1), (c) preference for P 1 without switching (ρ 1 > ρ 2 , α = 1), (d) preference for P 1 with switching (ρ 1 > ρ 2 , α > 1), (e) preference for P 2 without switching (ρ 1 < ρ 2 , α = 1), (f) preference for P 2 with switching (ρ 1 < ρ 2 , α > 1).
The diamond-shaped food web model 106 Switching and preference can have dramatic effects on the dynamics of plankton food 107 web models. As a relatively simple example, consider the following diamond-shaped 108 food web with one nutrient resource (N ), two competing phytoplankton types (P 1 and 109 P 2 ), and a single zooplankton species (Z) that feeds upon both phytoplankton. The We assume that zooplankton are relatively strong swimmers and are not 115 mixed out. The maximum growth rates of P 1 and P 2 are given by µ 1 and µ 2 . Nutrient 116 uptake by the phytoplankton follows Monod kinetics with half-saturation constants k 1 117 and k 2 . The phytoplankton non-grazing mortality rate is given by m p and the mortality 118 of the zooplankton is given by m z . γ is the growth efficiency of the zooplankton. The 119 dynamics of this food web are given by Coexistence dynamics in the diamond-shaped food web model

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In the absence of any grazing (i.e., when Z = 0), the phytoplankton type that is the 122 better competitor for nutrients will eventually exclude the other phytoplankton type in 123 model (2)-(5) [33]. We define the value as a metric for the competitive ability of a phytoplankton type. If λ i < λ j , then P i is a 125 stronger competitor for nutrients than P j and P i will drive P j to extinction [33]. We 126 chose µ 1 > µ 2 and k 1 < k 2 to satisfy the criteria that P 1 is the stronger competitor 127 (Table 1). Zooplankton preference and switching may allow for coexistence between P 1 128 and P 2 in situations where, in the absence of the zooplankton, P 1 would drive P 2 to extinction. Consider the scenario where P 2 = 0; the equilibrium point of the linear food 130 chain composed of N , P 1 , and Z is given by Near this equilibrium point, the dynamics of P 2 are given by the linearization of model 132 (2)-(5). The population of type P 2 will grow (or shrink) according to the invasion 133 growth rate r-the per capita growth rate of P 2 when P 2 ≈ 0: If r > 0, then P 2 can successfully invade the system.

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Note that the term ρ2gmaxZ * ksat+ρ1P * 1 in Eq 10 is always positive, so the growth of a 136 phytoplankton type when its density is near zero is strictly greater when the 137 zooplankton grazer displays switching. Switching behavior, therefore, is a kind of 138 density-dependent mortality that provides a refuge for phytoplankton at low densities. 139 Eq 10 suggests that there exist some values of ρ i and α such that r > 0 even when 140 λ 2 < λ 1 . Stated differently, a weaker phytoplankton competitor should be able to invade 141 the equilibrium of a stronger phytoplankton competitor under certain grazing conditions. 142 We explored this hypothesis further by simulating the diamond-shaped food web model 143 in each of six test cases which cover different combinations of preference and switching. 144 Case 1: control scenario, no preference (ρ 1 = ρ 2 = 1) and no switching (α = 1)

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For each case, we simulated model (2)-(5) over a range of input nutrient 151 concentrations (N 0 ) and mixing rates (D) using random initial conditions drawn from 152 uniform distributions. We observed four possible asymptotic outcomes: (1) P 1 drives P 2 153 to extinction, (2) P 2 drives P 1 to extinction, (3) P 1 and P 2 exist in a stable equilibrium, 154 or (4) P 1 and P 2 exist in an unstable equilibrium characterized by oscillations. For 155 these simulations, P 1 was the stronger competitor and P 2 was the weaker competitor.

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These simulations suggest that coexistence is possible in the diamond-shaped food 157 web model if (1) the zooplankton have a preference for the stronger competitor or (2) 158 the zooplankton exhibit switching (Fig 2). However, the range of environmental 159 conditions (i.e., nutrient inputs and mixing rates) that allows for coexistence in Cases 2 160 and 3, where zooplankton exhibit preference only, is quite small, indicating that 161 coexistence mediated by preference alone is rare and represents a delicate balance 162 between zooplankton preference and the growth rates and nutrient affinities of the 163 type to extinction (Fig 2). When zooplankton have a preference for the stronger 168 phytoplankton and display no switching (Case 2), P 2 > 0 and P 1 = 0 at equilibrium.

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The simulation results were consistent with our predictions based on Eq 10. The r = 0 170 line where the invasion growth rate for the weaker competitor changes from negative to 171 positive corresponds to the boundary between the region of the parameter space where 172 the model is asymptotically stable with P 2 = 0 and the region where stable coexistence 173 between P 1 and P 2 occurs (Fig 2). The KTW functional response has one characteristic that bears particular attention.

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The grazing rate on one phytoplankton type can increase as the density of an alternative 177 phytoplankton type increases (Fig 3). Here, we coin the term synergistic grazing to 178 describe this phenomenon. Mathematically, synergistic grazing occurs when dGi dPj > 0 for 179 i = j. Synergistic grazing is not unique to the KTW functional response [16]. Consider the scenario in which we begin with only one phytoplankton type (P 1 ) and 186 we calculate the grazing rate on P 1 as we introduce a secondary phytoplankton type 187 (P 2 ). The total grazing rate must increase because the total phytoplankton biomass is 188 increasing. However, if switching is strong (α >> 1), then the distribution of grazing 189 pressure will be heavily skewed towards P 1 since P 2 exists at low densities in the 190 environment. In this case, the grazing rate on P 1 will increase as P 2 is introduced even 191 though the density of P 1 is being held constant. Perhaps counterintuitively, synergistic 192 grazing has a stronger effect when the predator has a strong preference for the less 193 abundant phytoplankton type because the value of ρ i acts as a modifier of the "effective 194 biomass" in the system. Therefore, if ρ 2 is large, then more effective biomass is added 195 to the system when P 2 is introduced and the grazing rate changes faster than it would if 196 the zooplankton had a lower preference for P 2 .

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Synergistic grazing alters the dynamics of the diamond-shaped food web model in 198 interesting ways. It has a larger effect when switching behavior is stronger (Fig 3) and 199 in some cases can result in coexistence criteria that are counterintuitive. For example, 200 in a model with no switching, the invasion growth rate of P 2 will decrease as the 201 zooplankton preference for P 2 increases (Fig 3). However, when switching is included, 202 the invasion growth rate is independent of ρ 2 , meaning that P 2 will be able to invade 203 the system no matter how large the grazer preference for P 2 becomes. This occurs 204 because switching creates a grazing refuge for phytoplankton at low densities. Grazer 205 preference has no effect when a phytoplankton type is rare because all of the grazing is 206 focused on the more common phytoplankton type.

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The size-structured phytoplankton model 208 Phytoplankton communities rarely exist as only two types, but instead are commonly 209 composed of many different types coexisting simultaneously. Size is a particularly 210 important trait for structuring these communities since cell size is related to many 211 important characteristics of phytoplankton, including growth rates and nutrient 212 affinities [45]. To explore the effects of zooplankton preference and switching on 213 phytoplankton communities structured by cell size, we extended the model above to an 214 NPZ model that includes an arbitrary number of phytoplankton size classes. This new 215 model maintains a single zooplankton species that distributes grazing pressure across all 216 phytoplankton size classes according to the KTW functional response. The 217 size-structured model is written as The parameters µ i and k i are the growth rate and nutrient affinity for each 219 phytoplankton size class. We chose to scale these parameters allometrically (see Table 2) 220 such that the smallest phytoplankton size classes have the largest µ and the smallest k 221 following previous size-structured phytoplankton models [34,35]. As size increases, 222 competitive ability decreases. Other parameters used to simulate the model are similar 223 to those selected for the diamond-shaped model ( Table 2). The distribution of grazer preference between size classes is more complex in model 239 (11)- (13). For Case 7 (no preference), by definition, ρ i is equal across all size classes.

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For Case 8 (preference for smaller size classes), we set ρ i to decrease linearly across size 241 classes. For Case 9 (preference for larger size classes), we set ρ i to increase linearly 242 across size classes. Taking ρ 1 and ρ n as the preference for the largest and smallest size 243 classes, then ρ i for each size class is Previous analysis of the KTW functional response has assumed a constant 245 α = 2 [20]. With that same assumption, we find that coexistence between at least two 246 phytoplankton size classes is possible in Cases 7 and 8, as long as the influx water 247 nutrient concentration (N 0 ) is large enough (Fig 4). In Case 9, the smallest 248 phytoplankton size class dominated and excluded all other size classes. In all of these 249 simulations, the smallest phytoplankton size class is always the only one to exist by 250 itself at low nutrient input rates and larger size classes are added in order of increasing 251 size as N 0 increases (Fig 4). 252 We further explored how variable switching strength affected diversity by simulating 253 the size-structured model for α = 1 to 3. In these simulations, we also considered a 254 range of relative preference for smaller size classes (ρ 1 − ρ n ) where relative preference 255 Fig 4. Simulations of the size-structured phytoplankton community model. We simulated model (11)-(13) with ten phytoplankton size classes for 30 values of N 0 ranging from 1-15 mmolN m −3 . We chose initial conditions at random from uniform distributions (N (0) ∼ U [0, 15], P i (0) ∼ U [0, 2], Z(0) ∼ U [0, 10]), and simulated the model from t = 0 to t = 10, 000 days. Each bar represents the steady-state phytoplankton community for a given value of N 0 . The height of the bar gives the total phytoplankton biomass and the bar is color coded by the cell volume of a given size class. Case 7 includes switching without preference (ρ 1 = ρ 10 = 1, α = 2), Case 8 includes switching with preference for smaller size classes (ρ 1 = 1, ρ 10 = 0.5, α = 2), and Case 9 includes switching with preference for larger size classes (ρ 1 = 0.5, ρ 10 = 1, α = 2). equals zero corresponds to Case 7 and values greater than zero correspond to Case 8 256 (Fig 5). Phytoplankton diversity, as represented by the number of coexisting size classes 257 at steady state, is higher under stronger zooplankton switching and increased relative 258 preference for smaller phytoplankton size classes. For Case 9, where zooplankton have 259 preference for larger size classes, we found a counterintuitive relationship between 260 preference and its impacts on phytoplankton size structure; we treat this in more detail 261 in the next section. grazing are more evident when α is larger. Intuitively, we expect that increased 267 zooplankton preference for large size classes will make it harder for these size classes to 268 persist, since larger cells are weaker competitors for nutrients to begin with. However, 269 because of the effects of synergistic grazing in the KTW functional response, we actually 270 found that larger size classes benefited from increased grazer preference when switching 271 was very strong. 272 We ran a series of simulations of the size-structured model using α = 10 and 273 different values of relative preference for larger size classes (ρ n − ρ 1 ) where relative 274 preference equals zero corresponds to Case 7 and positive values correspond to Case 9 275 (Fig 6). Contrary to our expectation, an increased relative preference for larger size 276 classes resulted in higher biomass within the largest phytoplankton size class and lower 277 biomass in the smallest size class. The response in the intermediate size classes was 278 complex, resulting in some cases in non-monotonic changes in the size class biomass as 279 the relative preference was adjusted.

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Strong switching results in grazing pressure that is heavily skewed towards the 281 smaller size classes since they compose the majority of the phytoplankton biomass. This 282 creates a scenario in which the grazing pressure on large size classes is low (due to their 283 very low densities), even when there is strong preference for large size classes. Instead, 284 strong preference for larger size classes increases the total preference-weighted biomass 285 in the system, resulting in a higher overall grazing rate. A higher grazing rate actually 286 Fig 6. Synergistic grazing in the size-structured phytoplankton community model. Mean biomass concentration in each of the ten phytoplankton size classes over the steady-state limit cycle as a function of the difference between zooplankton preference for the largest and smallest size classes (ρ 10 − ρ 1 ). A larger difference indicates a stronger relative preference for larger phytoplankton size classes. Each line is labeled, with one referring to the smallest size class and ten referring to the largest size class. For these simulations, α = 10.
favors the larger size classes since strong switching ensures that most of the grazing is 287 focused on the smaller, more abundant phytoplankton types.

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Predator switching behavior is frequently observed in laboratory and field 290 studies [14,[36][37][38][39]; modeling work has shown that it stabilizes dynamics and increases 291 diversity [8,19,20,40]. Gentleman et al. [16] provide a thorough review of the functional 292 responses commonly used in biogeochemical models, many of which include grazer 293 preference and switching. Here, we analyzed the KTW functional response to evaluate 294 the mechanisms by which preference and switching allow coexistence under 295 circumstances that would otherwise lead to competitive exclusion. We have extended 296 the work of Vallina et al. [20] by examining how the functional response behaves on the 297 scale of competition between individual phytoplankton types. We have also identified 298 and discussed an important characteristic of the functional response, which we have 299 termed "synergistic grazing." Synergistic grazing can result in biologically 300 counterintuitive dynamics, particularly when strong switching is combined with 301 preference for weaker competitors. 302 We found that the KTW functional response allowed for grazer-mediated coexistence 303 through both grazer preference and switching as independent mechanisms. Switching 304 was generally the more powerful mechanism for generating coexistence, allowing for 305 coexistence between competing phytoplankton types across a broad set of environmental 306 conditions. In contrast, grazer preference in the absence of switching must be carefully 307 balanced with the competitive ability of the phytoplankton types to allow for 308 coexistence. When combining both zooplankton preference and switching in the KTW 309 parameterization, caution is warranted as the interaction can produce some unexpected 310 behaviors. Synergistic grazing is an example of a potentially problematic characteristic 311 that emerges as a result of the interaction between zooplankton preference and 312 switching. The effects of synergistic grazing are most evident when switching is strong 313 and when zooplankton have a preference for the weaker competitor. It is worth noting 314 that synergistic grazing arises directly from the fact that total grazing rate and the 315 distribution of the total grazing rate onto individual phytoplankton types are calculated 316 as independent terms. This functional form was chosen specifically to avoid the problem 317 of antagonistic grazing [20], another example of a problematic characteristic of grazing 318 functional responses. Therefore, it appears that there is a trade-off between functional 319 responses that display antagonistic grazing and the KTW functional response, which 320 does not display antagonistic grazing, but does include synergistic grazing. To date, we 321 are not aware of any functional response that includes a representation of switching 322 behavior and displays neither antagonistic grazing nor synergistic grazing [16]. 323 Conceptually, the switching functional response used in this model could represent 324 multiple ecological mechanisms that all have the same emergent behavior: 325 frequency-dependent differential grazing on multiple available phytoplankton types. The 326 classical view of switching is that it arises from zooplankton behavior such as changes in 327 November 13, 2019 10/14 feeding strategy or time budget optimization in response to a variable phytoplankton 328 community [41][42][43]. Alternatively, the zooplankton represented in this model could be 329 viewed as a generalized grazer community rather than a single species. Under this 330 scenario, the switching "behaviors" represent internal changes in the composition of the 331 zooplankton community as the composition of the phytoplankton community evolves 332 through time. When one phytoplankton type becomes very abundant, specialized 333 predators of that type also become more abundant and the integrated grazer community 334 becomes more efficient at grazing that phytoplankton type. Conversely, when a 335 phytoplankton type is rare, its specialized predators will also be rare and the proportion 336 of total grazing pressure on this phytoplankton type will be small. From either 337 perspective, the mathematical consequences are the same and so the parameterizations 338 used here could be used to represent either ecological scenario.

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Marine ecologists have long acknowledged the role that cell size plays in structuring 340 phytoplankton communities [44]. Observations from a variety of diverse biogeochemical 341 regimes have consistently shown an inverse relationship between cell size and 342 phytoplankton abundance [45]. Furthermore, the fractional abundance of small cells in 343 a phytoplankton community increases as total biomass decreases [45]. There has been 344 some debate among ecologists as to exactly what environmental factors control 345 phytoplankton community size structure, some claiming that nutrient supply alone is 346 sufficient to explain the variability [46,47], while others argue that temperature has an 347 important effect [48]. Our analysis highlights another layer in this debate; grazing 348 pressure, particularly when zooplankton display preference or switching, can play an 349 important role in structuring phytoplankton communities. The inclusion of preference 350 and switching behaviors increases phytoplankton diversity and restrains the dominance 351 of smaller cells that, generally speaking, are stronger competitors for nutrients.