Accepted Manuscript

Complex Daphnia interactions with parasites and competitors C.E. Caceres, G. Davis, S. Duple, S.R. Hall, A. Koss, L. Ping, ´ Z. Rapti PII: DOI: Reference:

S0025-5564(14)00216-8 10.1016/j.mbs.2014.10.002 MBS 7549

To appear in:

Mathematical Biosciences

Received date: Revised date: Accepted date:

8 November 2013 1 October 2014 7 October 2014

Please cite this article as: C.E. Caceres, G. Davis, S. Duple, S.R. Hall, A. Koss, L. Ping, Z. Rapti, ´ Complex Daphnia interactions with parasites and competitors, Mathematical Biosciences (2014), doi: 10.1016/j.mbs.2014.10.002

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• Diluters that are superior competitors wipe out the focal host. • Diluters that are inferior competitors may reduce the density of the focal hosts. • Diluters undermine the stabilizing effects of disease. • The negative effects of competition might outweigh the positive effects of dilution.

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COMPLEX DAPHNIA INTERACTIONS WITH PARASITES AND COMPETITORS ´ C. E. CACERES, G. DAVIS, S. DUPLE, S. HALL, A. KOSS, P. LEE AND Z. RAPTI∗

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Abstract. Species interactions can strongly influence the size and dynamics of epidemics in populations of focal hosts. The ”dilution effect” provides a particularly interesting type of interaction from a biological standpoint. Diluters - other host species which resist infection but remove environmentallydistributed propagules of parasites (spores) - should reduce disease prevalence in focal hosts. However, diluters and focal hosts may compete for shared resources. This combination of positive (dilution) and negative (competition) effects could greatly complicate, even undermine, the benefits of dilution and diluter species from the perspective of the focal host. Motivated by an example from the plankton (i.e., zooplankton hosts, a fungal parasite, and algal resources), we study a model of dilution and competition. Our model reveals a suite of five results: • A diluter that is a superior competitor wipes out the host, regardless of parasitism. Although expected, this outcome is an ever-present danger in strategies that might use diluters to control disease. • If the diluter is an inferior competitor, it can reduce disease prevalence, despite the competition, as parameterized in our model. However, competition may also reduce density of susceptible hosts to levels below that seen in focal host-parasite systems alone. • As they decrease disease prevalence, diluters destabilize dynamics of the focal host and their resources. Thus, diluters undermine the stabilizing effects of disease. • The four species combination can generate very complex dynamics, including period-doubling bifurcations and torus (Neimark-Sacker) bifurcations. • At lower resource carrying capacity, the diluter’s dilution of spores is ’helpful’ to the focal host, i.e., dilution can elevate host density by reducing disease. But, as the resource carrying capacity increases further, the equilibrium density of the diluter increases while the density of the focal host decreases, despite competition. Namely, the negative effects of competition start to outweigh the positive effects of dilution from the perspective of equilibrium density of the focal host.

1. Introduction

The dynamics of host-pathogen interactions often hinge on other species [36], [37], [38], [41], [53]. For instance, predators can either inhibit or facilitate disease spread, depending on how they interact with both host and parasite [6], [7], [13],

Date: October 21, 2014. 1991 Mathematics Subject Classification. Primary: 92D30; Secondary, 37N25 (37G15). Key words and phrases. Daphnia, epidemic model, Holling model, bifurcation theory, diseasefree equilibria, endemic equilibria, resource competition. ∗ Corresponding author, Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, IL 61801, Tel: 217-333-6625, Fax: 217-333-9576, email: [email protected]. 2

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[18], [27], [31], [48], [50]. Additionally, other potential host species vary in their susceptibility to parasites. These hosts can also alter disease dynamics. When they act as highly competent hosts or support large populations of vectors, disease prevalence in the focal host may increase [46], [52], [59]. In contrast, other species serve as poor or completely resistant hosts. When these species reduce contact between focal hosts and parasites (through consumption of environmentally-distributed spores or ’wasted bites’ of vectors), these players can reduce disease via a ”dilution effect” [2], [12], [49], [57]. Thus, species interactions can strongly shape epidemiology. However, these interactions can become complex and difficult to predict [26], [35], [39], [40]. We must continue developing theory at this interface of community ecology and disease. Here, we focus on competition-dilution, a type of interaction between hosts that compete for a shared resource [20]. In this interaction, the focal host is more vulnerable to a parasite that spreads via environmentally distributed propagules (hereafter: spores). (This kind of epidemiology is quite common in non-human systems, e.g., snail-worm, mammal-worm, insect-virus, bacteria-virus, etc.; [14]). The second host is highly resistant - even completely resistant - to infection. Importantly, this resistant host unselectively removes spores from host habitat while feeding. Since it acts as a ”diluter”, it should decrease prevalence of infection in the focal host [30], [37], [38]. The diluter, then should indirectly facilitate the focal host (by enhancing its fitness). However, both hosts also compete for a shared resource [16], [58]. Resource competition introduces a negative fitness component for the focal host and may greatly complicate the dynamical interplay between the two hosts and their parasite. Our study delineates some of the dynamical outcomes possible in competition-dilution. Once resource competition becomes intertwined with the dilution effect, three suites of issues arise. First, maintenance of diversity becomes uncertain. For instance, a diluter which is a superior competitor (i.e., has a lower minimal requirement for the shared resource) can displace the focal host, whether or not it is experiencing an epidemic [16]. In contrast, epidemics may facilitate coexistence of a diluter which is an inferior competitor - permanently or only transiently - if disease raises available resources to levels that support the diluter [4], [16], [26], [32]. Second, the ’success’ of dilution becomes less certain with competition-dilution. Successful dilution should reduce disease prevalence, but also may increase density of healthy (uninfected) hosts - a favorable outcome from a management perspective [37], [47]. Competition between hosts may undermine this last outcome in particular, since the competing diluter may depress density of focal hosts [3]. Such a result might undermine the appeal of the dilution effect for disease control. Third, dynamics of the four species system become less certain, particularly regarding oscillations. For instance, host-resource systems might have a tendency to oscillate when hosts overexploit their resources (as in the Rosenzweig-McArthur system; [44], [45]). Disease can stabilize those oscillations by imposing mortality on the host (and might even increase host density through hydra effect; [1], [28]). However, diluters must again destabilize these interactions by reducing disease (thereby decreasing a mortality factor for the focal host). All of these open-ended issues point to the pressing need for theory for competition-dilution. We tackle these issues with a model crafted around a motivating example. The focal host is a crustacean herbivore Daphnia dentifera [23]. Daphnia and their algal

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resource have generated a great deal of interest from a dynamical perspective [17], [44]. This host becomes infected by a fungal parasite (Metschnikowia bicuspidata) after incidentally consuming fungal spores distributed in the host’s habitat (the water column of lakes) while feeding [8], [19]. Thus, infection is tightly tied to feeding and therefore resource density (since feeding depends on resources; [19]. Infection, however, ultimately leads to death the hosts and release of a resource-dependent density of spores ([15], [21], [22]). Simultaneously, the focal host competes with other zooplankton, such as Daphnia pulicaria and Ceriodaphnia spp., for those algal resources. Plankton provides excellent examples of competition [5], [9], [42]. These competing taxa, however, act as diluters - they consume spores while having very low susceptibility. Dilution may explain why density of e.g., D. pulicaria correlates with the size and start of epidemics [20], [23], [51]. Although, given the complexity of the combination of dilution and competition, this correlation might not be so straightforward. Thus, the natural history of this system exhibits the key ingredients of competition-dilution. In section 2, we outline the model for the five populations. In section 3 we indicate conditions for the positivity and stability of the biologically feasible equilibrium points. In section 4 we study the bifurcations that the system undergoes. A discussion of the biological implications of our results comprises section 5. In the Appendix we present details of the mathematical analysis. 2. Model

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We divide the focal host (Daphnia dentifera) population into the susceptible (S) and the infected (I) class. We denote the fungal spores by Z, algae by A, and the competitor (Daphnia pulicaria) by D. Then, the dynamical system is written as follows: fS (A) SZ A

(1)

dS dt

= eS fS (A)(S + ρI) − (d + pS )S − µ

(2)

dI dt

= µ

dZ dt

= σeS fS (A)[d + v]I − λZ − [fS (A)S + fS (A)I + fD (A)D]

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(3)

fS (A) SZ − [d + v + pI ]I A

A )A − [fS (A)S + fS (A)I + fD (A)D] K

(4)

dA dt

= r(1 −

(5)

dD dt

= eD fD (A)D − (dD + pD )D.

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The algae dependent feeding rates for the two Daphnia species are assumed to be of type II [29]

(6)

fS (A) = fI (A) =

fS0 A fD0 A and fD (A) = . hS + A hD + A

All five population densities have been transformed to mg C/L and all parameters are strictly positive.

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Table 1. Variables and parameters

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Unit mg C/L mg C/L mg C/L mg C/L mg C/L day Value 0.6 mg C/ mg C [56] 0.32, 0.28/day mg C/mg C [56] 0.05-0.90 mg C/L [56] 0.5 mg C/L [56] 0.9 [24] 0.1/day, 0.3/day [56] 0.03/day [20] 10 mg C/mg C [24] 0.05/day [24] 31 days × mg C / mg C [20] 0.2/day [20] 0.2/day [56] 0.00-6.0 mg C/L (large range)

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Variable S susceptible host I infected host Z fungal spores A algae D competitor t time Parameter eS = eD : conversion efficiency fS0 , fD0 : maximal feeding rate hS : half saturation constant (susceptible) hD : half saturation constant (competitor) ρ : fecundity reduction due to infection pS = pD , pI : predation rate d = dD : background mortality µ : per spore infectivity v : virulence σ : spore release parameter λ : spore loss rate r : algal net maximal growth rate K : algal carrying capacity

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The susceptible class S (1) increases due to births from susceptible and infected hosts I, since disease in this system is transmitted only horizontally. Although, in the latter class births occur at a reduced rate due to the infection (0 ≤ ρ < 1). Growth in both species occurs by conversion of consumed algae. Hence, the birth rate depends on the feeding rate fS (A) and is proportional to the conversion efficiency eS of the consumed algal resources into Daphnia biomass. Susceptible Daphnia die at a constant background rate d and are preyed upon at a constant rate pS . Predator rate is treated as a parameter, not a variable, in our model due to the much slower timescale of predator dynamics. But this mortality rate can vary among the three host classes. Susceptibles move into the infected class after being successfully infected by the parasite as governed by the transmission rate. The transmission rate, in turn, depends on host density S, the host’s feeding rate fS (A), the relative density of spores to algal resources Z/A, and the per spore infectivity µ. We assume that Daphnia do not feed selectively, hence the spores experience the same risk of being eaten as the algae. The death rate of infected individuals (2) is increased by v due to the infection. Also, infected hosts are being preyed upon more intensely at a rate pI > pS due to selective predation. Predation on the infected class is higher because visually oriented predators selectively cull the more opaque (conspicuous) infected hosts. Spore production (spore yield upon death of infected hosts) increases with host growth rate [20, 24, 25]. Hence, spores Z (3) are released in the water column at a rate proportional to the per capita yield of spores σ(A) = σeS fS (A) and the death rate of infected hosts (d + v)I. Spores are removed from the water column when

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3. Equilibrium points

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consumed by Daphnia at a rate proportional to their feeding rate or by other causes (e.g., sinking, UV radiation) at a rate λ. Algae (4) grows logistically in the absence of any Daphnia at an intrinsic rate r and with carrying capacity K. It is consumed by all three classes of Daphnia. The competitor species D (5) increases due to births at a rate proportional to its feeding rate fD (A) and its conversion efficiency eD of the consumed algal resources. Their background mortality occurs at a rate dD and they are being preyed upon at a rate pD . We assume that the Daphnia get no nutritional benefit from eating spores.

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Several assumptions must hold for the equilibrium points that we will focus on. First, the only feasible equilibrium points are the ones with nonnegative population densities. Second, if the equilibrium value for the algae is zero, the equilibrium values for the remaining populations must be zero too. In other words, without algal resources, the populations that feed on them, also die out. Another restriction is that the spore and infected class densities are either both zero, or both positive. This assumption is necessary to model the disease dynamics realistically: spores are necessary to infect the susceptible class, and without the infected host, the parasite dies out. The former case corresponds to a disease-free equilibrium, namely a steady-state solution where there is no disease, while the latter one corresponds to an endemic equilibrium point: a steady-state solution where the disease persists. Finally, the infected class can persist only when the susceptible class is present too. Hence, when S ∗ = 0, we also need I ∗ = Z ∗ = 0. We will use ∗ to annotate equilibrium values. The following lemma guarantees that the subset O = {(S, I, Z, A, D)|S ≥ 0, I ≥ 0, Z ≥ 0, A ≥ 0, D ≥ 0} ⊂ R5 is forward invariant for our system. Lemma 1. The positive orthant in five dimensions is forward invariant for the system of differential equations (1-5).

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Proof. From Eq. (1) it follows that S = 0 ⇒ dS dt ≥ 0 for (S, I, Z, A, D) ∈ O. dZ Similarly, I = 0 ⇒ dI ≥ 0 and Z = 0 ⇒ ≥ 0 for (S, I, Z, A, D) ∈ O. Eq. dt dt (4) and (5), respectively, yield A = 0 ⇒ dA = 0 and D = 0 ⇒ dD dt dt = 0, for (S, I, Z, A, D) ∈ O.  The lemma that follows, guarantees that in O, all equilibrium points are feasible.

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Lemma 2. (i) A∗ = 0 implies S ∗ = I ∗ = Z ∗ = D∗ = 0; (ii) S ∗ = 0 implies I ∗ = Z ∗ = 0; (iii) I ∗ = 0 if and only if Z ∗ = 0.

Proof. (i) For an equilibrium with A∗ = 0 which implies dA dt = 0, it follows from (1) that S ∗ = 0, from (2) that I ∗ = 0, from (3) that Z ∗ = 0, and from (5) that D∗ = 0. (ii) If S ∗ = 0, then (2) implies I ∗ = 0, which in turn combined with (3) implies Z ∗ = 0. (iii) If I ∗ = 0, then (2) yields Z ∗ = 0, since we are in O and the parameters of the system are positive. Conversely, if Z ∗ = 0, then (2) implies I ∗ = 0.  3.1. Existence.

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3.1.1. Trivial and Resource-only equilibria. By inspection one can find the following two equilibria in O: the trivial equilibrium E0 = (0, 0, 0, 0, 0) and the equilibrium where only algae persists and equals its carrying capacity EA = (0, 0, 0, K, 0). Both of these exist for all parameter values.

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=

(8)

∗ DD

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This equilibrium is in O if (9)

hD (dD + pD ) eD fD0 − (dD + pD ) reD A∗D (K − A∗D ) . K(dD + pD )

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3.1.2. Competitor-Resource equilibrium. From Lemma 2, it follows that, another equilibrium is the one where S ∗ = 0, and only algae and the competitor persist ∗ ∗ ED = (0, 0, 0, A∗D , DD ). The values for the equilibrium densities A∗D , DD are found in Appendix A to be

eD fD0 − (dD + pD ) > 0

K ≥ A∗D .

(10)

Condition (9) states that the death rate of the competitor cannot exceed the maximal birth rate and condition (10) states that the system is enriched enough to meet the minimal resource requirement of the competitor. When A∗D = K , it holds ∗ = 0 and this equilibrium coalesces with EA . DD

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3.1.3. Host-Resource (disease-free) equilibrium. In the absence of disease I ∗ = Z ∗ = 0, besides the three equilibria Ei , i = 0, A, D described previously, we also have the equilibrium where algae and susceptible host persist ES = (SS∗ , 0, 0, A∗S , 0). The values for the equilibrium densities SS∗ , A∗S are found in Appendix A to be (11)

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(12)

This equilibrium is in O if (13)

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eS fS0 − (d + pS ) > 0

K ≥ A∗S .

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Condition (13) states that the death rate of the host cannot exceed the maximal birth rate and condition (14) states that the system is enriched enough to meet the minimal resource requirement of the host. When A∗S = K , it holds SS∗ = 0 and this equilibrium coalesces with EA . Before we study the remaining equilibria, we will calculate the basic reproduction number R0 for our model. It is defined as the expected number of secondary cases produced, in a completely susceptible population, by a typical infective individual [10]. It is also a re-expression of the parasite’s ability to invade into the resource-host subsystem. To calculate R0 in our case, where the free living spores cannot maintain themselves through growth in the water column, we adopt the next generation approach [10, 60]. According to this theory, R0 is the spectral radius of the matrix fv−1 , where f is the transmission matrix describing the generation of secondary infections in the I, Z compartments, while v is the transition matrix describing the

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and R0 =

(15)

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or equivalently

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transfer of individuals by all other means into/out of the I, Z compartments of our model. It follows that ! f (A∗ ) 0 µ SA∗ S SS∗ S f = 0 0 ! d + v + pI 0 v = f (A∗ ) −σeS fS (A∗S )(d + v) λ + SA∗ S SS∗

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eS fS0 − 1 + hKS (d + pS ) rσµ(d + v)(d + pS )
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Equation (15) states that the basic reproduction number is proportional to the f (A∗ ) transmission rate per infective propagule µ SA∗ S SS∗ , the rate of release of propagS ules per dead infected individual σeS fS (A∗S )(d+v), the life span of the infected class  −1 f (A∗ ) (d+v+pI )−1 , and the life span of the propagules λ + SA∗ S SS∗ , which includes S the background spore loss λ and the consumption by the susceptible Daphnia. In figure 1 we show the dependence of the equilibrium densities of A and S and of R0 on K (algal carrying capacity) and hS (focal host half-saturation constant) in the case when the disease free equilibrium persists. As K increases, the algal equilibrium density is fixed, and the host equilibrium density increases. Hence, in equation (15) all terms except SS∗ are fixed. This means that as K increases it is easier for the parasite to invade, since there are more hosts to infect. As hS increases, the host becomes a less effective consumer (since fS (A) decreases), so its equilibrium density SS∗ decreases and A∗S increases. This, in turn, implies that the spore yield rate and the life span of the parasite increase, while the transmission rate decreases. The net result for this model is that as hS increases, it is harder for the parasite to invade, because there are not enough susceptible hosts and because transmission is harder to occur. This is an interesting finding, since the spore yield increase could have outweighed the depressing effects of SS∗ and the transmission rate. In figure 1 we show this dependence for the the case when the the parasite cannot invade, namely R0 < 1.

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3.1.4. Host-Competitor-Resource (disease-free) equilibrium. Another type of equilibrium points with I ∗ = Z ∗ = 0 emerge if A∗D = A∗S . This equilibrium emerges when the two competitors have the same minimal resource requirements, namely when they are neutrally competitive. The possibility that this occurs is vanishingly small though: the set of points in the parameter space where this occurs is of measure zero. In this case, it must hold (16)

∗ ∗ ∗ fS0 SSD fD0 DSD A∗ fS0 SS∗ fD0 DD + = r(1 − SD ) = = . ∗ ∗ ∗ hS + AS hD + AD K hS + AS hD + A∗D

∗ ∗ We denote these equilibria by ESD = (SSD , 0, 0, A∗SD = A∗D = A∗S , DSD ). It follows ∗ ∗ ∗ ∗ ∗ ∗ ∗ from (16) that SS ≥ SSD and DD ≥ DSD . If SSD = SS then DSD = 0, so ESD

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Figure 1. In the top panels, we show how equilibrium values of algal resource A∗S and the susceptible host density SS∗ and also the reproductive ratio R0 , vary with carrying capacity K at a half saturation constant hs = 0.7. In the bottom panels, we show how A∗S , SS∗ and R0 vary with hS at a fixed K = 1.7. These are shown in the subsystem where the parasite cannot invade. All other parameters follow Table 1

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∗ ∗ ∗ merges with ES , while if DSD = DD then SSD = 0, so ESD merges with ED . As expected, ESD belongs to O, if

(17)

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K ≥ A∗S = A∗D .

∗ ∗ If A∗S = A∗D = K , it holds SSD = 0 = DSD and these equilibrium points coalesce with EA .

3.1.5. Host-Resource endemic equilibrium. In the case of endemic equilibrium points, it must hold S ∗ > 0, I ∗ > 0, Z ∗ > 0. The competitor might not (D∗ = 0) or might (D∗ > 0) persist. In the former case, it is shown in Appendix A that the equilibrium value for the algae A∗ satisfies a cubic polynomial, and the equilibrium point

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densities are = A∗

∗ SSIZA

=

∗ (21) ZSIZA

=

∗ (22) ISIZA

=

(20)

 r 1−

A∗ K



A∗ λ + r(1 − (d + v + pI )A 1 = µfS (A∗ ) σeS fS (A∗ )(d + v) R(A∗ ) fS (A∗ ) ∗ ∗ ∗ (d + v + pI )A eS (fS (A ) − fS (AS )) µfS (A∗ ) d + v + pI − ρeS fS (A∗ )   A∗ ∗ 1 r 1 − K A eS (fS (A∗ ) − fS (A∗S )) , R(A∗ ) fS (A∗ ) d + v + pI − ρeS fS (A∗ ) A∗ K )

∗

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(23)

σeS fS (A)(d + v)µr 1 − (λ + r(1 −

A K ))(d

+ v + pI )

It follows from (23) and (15) that when A = A∗S , the two expressions agree, namely R(A∗S ) = R0 . ∗ ∗ ∗ , A∗SIZA , 0). Then, , ZSIZA , ISIZA We denote this equilibrium by ESIZA = (SSIZA ESIZA ∈ O if

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(24)

∗

A∗S

A∗ ≥ 0,

A − ≥ 0. d + v + pI − ρeS fS (A∗ )

(25)

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(26)

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The second criterion (25) states that when the death rate of the infected hosts (d + v + pI ) exceeds their birth rate, then A∗ > A∗S . The dependence on K and hS of the equilibrium densities and the prevalence at ∗ equilibrium, defined as S ∗I+I ∗ is shown in figure 2. ∗ ∗ . Hence the criterion = 0 = ISIZA It can be seen that A∗ = A∗S implies ZSIZA ∗ (24) is not really restrictive. Also, A satisfies the identity R(A∗ ) = 1 +

eS (fS (A∗ ) − fS (A∗S )) . d + v + pI − pS − ρeS fS (A∗ )

AC

Hence, for A∗ = A∗S it follows that R(A∗S ) = 1 = R0 and A∗

∗ SSIZA

r(1 − KS )A∗S = = SS∗ . fS (A∗S )

Therefore, the equilibrium coalesces with ES , i.e., the parasite cannot invade and persist within the host-resource system.

3.1.6. Host-Competitor-Resource endemic equilibrium. Finally, in the case when the competitor persists (D∗ > 0), it can be shown (see Appendix A) that the

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Figure 2. The top six panels show the equilibrium densities and prevalence as K varies within the SIZA region for hS = 0.5. The bottom six panel show the equilibrium densities and prevalence as hS varies, for K = 1.8. All other parameters follow Table 1.

equilibrium values for the population densities satisfy the equations

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(27)

A∗SIZAD

= A∗D

(28)

∗ SSIZAD

=

(29)

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(30)

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(31)

∗ DSIZAD

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∗



A A∗ r 1 − KD A∗D (d + v + pI )A∗D λ + r(1 − KD ) = µfS (A∗D ) σeS fS (A∗D )(d + v) R(A∗D )fS (A∗D ) (d + v + pI )A∗D eS (fS (A∗D ) − fS (A∗S )) µfS (A∗D ) d + v + pI − ρeS fS (A∗D )   A∗ r 1 − KD A∗D e (f (A∗ ) − f (A∗ )) 1 S S S D S . R(A∗D ) fS (A∗D ) d + v + pI − ρeS fS (A∗D ) fS (A∗D ) ∗ ∗ ∗ DD − (S + ISIZAD ). fD (A∗D ) SIZAD

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∗ DD

eD fD0 − (dD + pD ) > 0, A∗D − A∗S ≥0 d + v + pI − ρeS fS (A∗D )   λ A∗D ≤ +1 K r fS (A∗D ) ∗ ∗ (S ≥ + ISIZAD ). fD (A∗D ) SIZAD

CR IP T

∗ ∗ ∗ ∗ This equilibrium which will be denoted by ESIZAD = (SSIZAD , ISIZAD , ZSIZAD , A∗SIZAD , DSIZAD ), is in O if

PT

ED

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The previous expressions state that the equilibrium algal density is set at the minimal resource requirement of the competitor A∗D . Now, algal density no longer increases with carrying capacity. In the host only system, A∗S was independent of K. With the addition of the infected class and the parasite A∗SIZA increased with K. Once the competitor is present, A∗ = A∗D is again independent of K. When the death rate d + v + pI of the infected host I exceeds its contribution to the birth rate ρeS fS (A∗D ), then A∗D has to be higher than the minimal resource requirement of the host A∗S . Also, clearly the competitor is less dense than it would be in a competitor-algae subsystem. ∗ In figures 3-4, we plot the equilibrium densities and the prevalence I ∗I+S ∗ in the SIZAD system and the system where the diluter is not present, as K and hS vary, respectively. Addition of a new species, the diluter-competitor, has two consequences. First, this species clears (removes) propagules. Second, it regulates density of the infected class - more specifically, it decreases it (see also [37] for a review on the dilution effect). Disease prevalence decreases, but resource competition also regulates host densities. In particular, at lower K the susceptible host density S ∗ is higher with the competitor D present, while at higher K the opposite is true. ∗ ∗ and the equilibrium coalesces with ESD . = 0 = ISIZAD If A∗D = A∗S then ZSIZAD In other words, the SIZAD system requires that the minimal resource requirement of the competitor also exceeds that of the focal host without the disease. Also, from (26) it follows that if A∗D = A∗ , then

AC

CE

∗ ∗ SSIZAD + ISIZAD

(32)

∗ DSIZAD

A∗ A∗ r(1 − ) and ∗ fS (A ) K fS (A∗ ) ∗ ∗ ∗ = DD − (S + ISIZAD ) fD (A∗ ) SIZAD A∗ A∗ ∗ = DD − r(1 − ) = 0. ∗ fD (A ) K

=

Hence, if A∗D = A∗ then the equilibrium coalesces with ESIZA . We now investigate how the host feeding traits and algal resources affect the disease dynamics. In particular, Lemma 3. In the case of stable equilibrium dynamics, when the competitor persists, disease prevalence is independent of the algal carrying capacity and decreases with increasing host half-saturation constant.

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Figure 3. Equilibrium densities and prevalence as K varies, in the system with and without the diluter present.

AC

Proof. Disease prevalence is defined as

(33)

p(t) =

I(t) . S(t) + I(t)

In the case of the ESIZAD equilibrium the prevalence (after some initial transient behavior) is constant in time and equals p(t) ≡

eS fS0 A∗D − (d + pS )(hS + A∗D ) , (v + pI − pS )(hS + A∗D ) + (1 − ρ)eS fS0 A∗D

where A∗D is independent of hS , K. The result follows readily since as hS increases, the numerator decreases and the denominator increases. This can also be seen graphically in figure 4. At the default values hS = 0.5 and K = 3 the disease

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Figure 4. Equilibrium densities and prevalence as the host’s halfsaturation constant hS varies, in the system with and without the diluter present.

AC

prevalence is 0.070 and for hS = 0.8 the prevalence drops to 0.003. At the same time, the equilibrium value for the susceptible class of the host population density equals   rA∗D (d + v + pI )(hS + A∗D )2 . S∗ = λ + r − 2 (d + v)A∗ K σµeS fS0 D Hence, S ∗ increases as K or hS increases.  Based on the previous lemma and figures 3 and 4, we notice the following. In the SIZAD system, the algal equilibrium density is set at the minimum resource requirement of the diluter A∗D and is independent of K. The same is true for the fungal spore equilibrium density Z ∗ . Instead, S ∗ , I ∗ , and D∗ all increase with

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K. Therefore, as K increases, D∗ also increases and clears up parasite propagules (decreases Z ∗ ) helping S ∗ increase too, along with I ∗ . The latter can be explained by the fact that since D∗ consumes spores, a larger infected class I ∗ is required to keep Z ∗ at a fixed value. Comparing the cases with and without the competitor, we see that in the absence of the competitor, Z ∗ would increase with K and I ∗ would increase more than it does when the competitor is present. One notices the interesting fact that for relatively low values of K, S ∗ is higher when the competitor is present. The opposite is true for higher values of K. This can be interpreted if one considers the competitive ability of the two hosts: when there are enough resources for both hosts, the incompetent one wins, although it is the inferior competitor. The total focal host population though, is (almost for all parameter values) higher when the competitor is not present. Namely, the competitor keeps both the infected class and the total focal host population at a lower equilibrium value. As hS increases, the equilibrium value of the susceptible host class also increases, as does the total focal host population, while the infected class, competitor, and propagule equilibrium densities decrease. For relatively high values of hS , S ∗ is higher in the presence of the competitor, than without it, although the competent host is now less efficient (hS > 0.5 = hD ).

M

3.2. Stability. The linear stability properties of equilibrium points Ei , i = 0, A, D, S can be obtained analytically. The equilibrium points ESD are non-hyperbolic, and the stability properties of equilibrium point ESIZA have only partially been obtained due to analytical intractability. The properties of ESIZAD are not easily analytically obtained. The following Theorem holds.

PT

ED

Theorem 4. (i) The trivial equilibrium E0 is a saddle for all parameter values. (ii) The equilibrium EA is linearly asymptotically stable if K < A∗S and K < A∗D . Otherwise, it is always a saddle. (iii) The equilibrium ED is linearly asymptotically stable if A∗D < A∗S (the incompetent host is a superior competitor to the focal host) and K − hD − 2A∗D < 0 (Hopf bifurcation). (iv) The equilibrium ES is linearly asymptotically stable if A∗S < A∗D < K (the focal host is a superior competitor to the incompetent host) and R0 < 1 (the parasite cannot invade). (v) The equilibrium ESIZA is linearly unstable if A∗SIZA > A∗D (when the host-disease system meets the minimal resource requirement of the incompetent host).

CE

Proof. The proof follows by the standard linear stability analysis performed in Appendix B. 

AC

The trivial equilibrium E0 is generically unstable for all consumer-resource models with this kind of structure, namely, logistic growth for the resource. The equilibrium EA is stable when the system is not enriched enough to support the consumers, i.e., their minimal requirements are not met. The equilibrium ED is stable when the diluter becomes the superior resource competitor, namely, it has a lower minimal resource requirement (figure 5). When K = A∗D the ED and EA equilibrium points undergo a transcritical bifurcation. The equilibrium ES is stable if the susceptible host, when not infected, has a lower minimal resource requirement than its competitor, but the system is not enriched enough to support the parasite yet (figure 6). When K = A∗S the ES and EA equilibrium points undergo a transcritical bifurcation. The ESA equilibrium becomes unstable when the minimal resource requirement of the diluter is smaller than the resource levels needed to support the SIZA subsystem (figure 7). The ESIZA equilibrium loses stability

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when A∗SIZA = A∗D in which case the ESIZAD equilibrium becomes stable (figure 8). This in turn loses stability through a Hopf bifurcation. Representative dynamics are shown in figure 9. All simulations of the five-dimensional ODE system (1-5), such as those in figures 5-9, were performed in Matlab [43], using the ode45 routine. Remark 1: Although we do not have an analytical proof, numerical results suggest the following. In the case of limit cycle dynamics, the maximum and amplitude of disease prevalence decrease with decreasing algal carrying capacity and increasing host half-saturation constant. The period-averaged prevalence experiences the same trend: it decreases/increases as K decreases/ hs increases. In the next section, we show the regions with different dynamical behavior in a bifurcation diagram.

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Figure 5. We show the dynamics when the system reaches the EDA equilibrium (K = 3 and hS = 0.85). The incompetent host D, which is the superior competitor, and the algae A persist, while the infected hosts I, the spores Z, and the focal host S do not. Note the differences in the time scales.

4. Bifurcations

Resource acquisition and use among hosts influences disease dynamics [22], [24]. Hence, we choose as bifurcation parameters the algal carrying capacity K and the half- saturation constant for the host hS (as hS increases, the host acquires the resource less efficiently). Both are expressed in mgC/L. The rest of the parameters are at the default values shown in table 1. Varying different parameters, such as fS , hD , fD , generates qualitatively similar bifurcation diagrams. We refer to figure 10 where various types of equilibria and cyclic behavior are shown. For values of hS exceeding the critical value hcr S = 0.8158, the focal host

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Figure 6. We show the dynamics when the system reaches the ESA equilibrium, namely when R0 < 1 (K = 1.5 and hS = 0.68). The susceptible host S, which is the superior competitor, and the algae A persist, while the infected hosts I, the spores Z, and the competitor D do not. Note the differences in the time scales.

AC

CE

PT

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(D. dentifera S) becomes the inferior competitor and gets outcompeted by the incompetent host (D. pulicaria D). In this case, the surviving populations A and D on the top region of figure 10, exhibit behavior mirroring that of the classic Rosenzweig - MacArthur model [55]. Specifically, for relatively low algal carrying capacity K < 1.7105 = A∗D only algae A persists (region annotated as A). As the branch point value K = A∗D is crossed, the equilibrium where algae and the diluter D persist becomes positive and asymptotically stable (region annotated as AD) and the equilibrium EA becomes linearly unstable. As we increase K further, the system undergoes a Hopf bifurcation causing A and D to undergo stable oscillations (region annotated as AD osc.). This phenomenon was observed in [54] in models of two-species ecosystems and is referred to as the paradox of enrichment: increasing the food or energy supply destroys the stable state as the amplitude of the oscillations grows. The critical value for K where the Hopf bifurcation occurs is K = hD + 2A∗D = 3.921. When the horizontal line hS = 0.8158 is crossed, and as long as K > AS and (34)

K
0 and Z > 0) giving rise to the

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Figure 7. We show the dynamics when the system reaches the ESIZA equilibrium with K = 1.5 and hS = 0.5.

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asymptotically stable equilibrium ESIZA (region annotated as SIZA). This remains stable until another branch point A∗SIZA = A∗D after which the incompetent host (D) persists too (region annotated as SIZAD). As K increases, A∗SIZA increases ∗ ∗ ∗ . Hence more infection by parasites that kills , ZSIZA , ISIZA too, and so do SSIZA hosts translates into a larger resource requirement for the support of the SIZA system, as was shown in figure 2. Eventually, with large enough K, the minimal resource requirement of the competitor D is met. This equilibrium ESIZAD is stable until the first Hopf bifurcation curve is crossed. This is reminiscent of the paradox of enrichment: destabilization of the full system as K increases. Actually, same as in the original paradox of enrichment case [54], the equilibrium algal density stays constant at A∗D as K increases, until the Hopf bifurcation occurs. Also, SIZAD oscillations become more probable as hS increases. The very effective consumer S is more likely to trigger oscillations because it is more likely to overexploit its resource and because the transmission rate decreases (as hS increases). This is evident from the slope of the Hopf curve in figure 10. There is a second Hopf curve for even larger values of K. The top branch is unstable, and as hS decreases, a generalized Hopf point occurs. The bottom branch is a stable Hopf curve. From the generalized Hopf point (annotated as Generalized Hopf in figure 10) emanates a limit point of cycles curve. Using the numerical continuation analysis software Matcont [11] and numerical simulations performed in MATLAB [43], period doubling bifurcations were found. For instance, at the point K = 3.5440377, hS = 0.5, the period of the cycles for all five populations densities increases from about 97 to 194 days. Our simulated periods are comparable to periods observed in disease-free microcosm experiments

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Figure 8. We show the dynamics when the system reaches the ESIZAD equilibrium with K = 3.0 and hS = 0.5.

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[17] in a microcosm experiment with Daphnia. Linear time series analysis of the data from a 2 year experiment revealed cyclic dynamics with periods ranging from 41 to 182 days. However, those experiments only included two Daphnia species, D. pulicaria and D. magna that were fed an assemblage of algae and other microorganisms, in the absence of parasites. In figure 11 we project the attractor to the three-dimensional space and show the susceptible host S ∗ , the infected host I ∗ , and incompetent host D∗ at a value of K before (3.543) and after (3.545) the period doubling bifurcation. The limit cycles are shown for the time ranges between 140,000 and 160,000 days. As the algal carrying capacity increases further, a series of period doubling bifurcations leads to a strange attractor as is shown in figure 12. All limit cycles in figure 12 are shown for the time ranges between 140,000 and 160,000 days. Such species oscillation and chaos are well known in planktonic resource competition models [56, 61] and have been used to offer insight in the paradox of the plankton [33, 34]. Following the Hopf curve in the SIZAD region in figure 10, as the host saturation constant hS and the algal carrying capacity K vary, Neimark-Sacker (torus) bifurcations take place. This is characterized by all populations oscillating with two frequencies: high-frequency oscillations with low-frequency modulation. An example of such oscillations is shown in figure 13. Data analysis in [17] suggested that a single fundamental frequency does not capture all the periodic behavior, and the authors conjectured that additional frequencies might be present. Hence, these torus bifurcations might capture this phenomenon. We also discovered regions where the SIZAD limit cycles collapse to either SA or SIZA limit cycles. In the former case, first the infected class I and the spores

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Figure 9. We show the dynamics after the ESIZAD equilibrium has undergone a Hopf bifurcation, K = 3.35 and hS = 0.45. For clarity we show the oscillations for large times and ignore the transient dynamics.

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Z get eliminated and then the competitor D get outcompeted by the focal host too. An example of such a limit cycle is shown in figure 14. In the latter case, the competitor dies out but the remaining populations undergo oscillations as those seen in figure 15. No sensitivity to initial conditions was observed anywhere else except in the region bounded by the Limit point cycle curve and the second Hopf curve in figure 10. Specifically, depending on the initial conditions either SA or SIZAD oscillations were observed. Finally, for comparison purposes, we show the bifurcation diagram corresponding to the figure 10 for the SIZA system in figure 16. One notices the absence of a Hopf curve -at least for 0 ≤ K ≤ 6. The only dynamically distinct regions are the ones where only algae A persists, only A and the focal host S persist, and the region of where the endemic equilibrium persists. 5. Discussion and conclusions

In this work, we studied a disease system with a focal host (D. dentifera), an incompetent host (D. pulicaria), infective free-living propagules (M. bicuspidata), and a common to all hosts food resource (algae). The focal host is assumed to be the superior competitor, while the incompetent host is the inferior competitor. As these two host classes feed on algae, they also consume infective propagules that exist in the water column. These are subsequently removed from and can only reenter the water column after the infected host dies releasing them. Given past studies [19]-[21] indicating the link between resource ecology and disease dynamics,

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Figure 10. Bifurcation diagram with varying hS and K. ”A” denotes the region where only the resources persist, ”AS” the algaefocal host region, ”AD” the algae-diluter region, ”SIZA” the endemic equilibrium where the diluter does not persist, ”SIZAD” the endemic equilibrium where the diluter persists, ”AD oscil.” the region where the diluter and the resource oscillate, which occurs after the Hopf line. Following the Hopf curve after the ”SIZAD” region, there is a period doubling bifurcation and a Neimark-Sacker bifurcation curve. There is also a Limit point cycle bifurcation and a second Hopf bifurcation curve which meet at the generalized Hopf point.

CE AC

Hopf

as bifurcation parameters we used the focal host’s half-saturation constant (hS ) and the algal carrying capacity (K) and investigated how these two parameters shape the epidemics.

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Figure 11. The left panel shows the projected stable limit cycle before the period doubling bifurcation at K = 3.543 and the right panel shows the projected stable limit cycle after the period doubling bifurcation at K = 3.545.

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Figure 12. We show the limit cycle at K = 3.91 (panel a), K = 3.93 (panel b), K = 3.94 (panel c), and K = 3.96 (panel d).

For high hS the behavior of the system is similar to the Holling system with algae as prey and the incompetent host as consumer. For small K only algae persists, and as K increases the incompetent host persists too, until the stable equilibrium is destroyed by a Hopf bifurcation. For lower values of hS the focal host persists, in a host-algae stable system. As either hS drops further – making

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Figure 13. The left panel shows the modulated oscillations of the infected host population and the right panel shows the modulated oscillations for the competitor for K = 2.9999 and hS = 0.133820. This corresponds to the torus-bifurcation region in the bifurcation diagram of figure 10.

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0.6

AC

Figure 14. We show the population densities of our system for K = 3.03 and hS = 0.1. The focal host S and the algae A undergo sustained oscillations, while the infected class, spores, and the competitor die out.

the host more efficient – or as K increases, the parasite can persist (in a stable manner), but not the competitor. In either case, the host equilibrium density increases driving up the basic reproductive ratio R0 . The equilibrium with the susceptible and infected host classes and the infective propagules persists until the minimal resource requirement for the inferior competitor is met leading to a stable equilibrium with all five populations present. This stable equilibrium is destructed, as K increases and hS decreases, by a Hopf bifurcation. Before the Hopf bifurcation, prevalence is independent of K since all three host classes increase, with

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´ C. E. CACERES, G. DAVIS, S. DUPLE, S. HALL, A. KOSS, P. LEE AND Z. RAPTI∗

1 0.5

5.55

0.4 0.2 0 5.5

5.6

time

4

0.8

3

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2 1 0 5.5

5.55

time

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x 10

D, mg C/L

A, mg C/L

0 5.5

I Z

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0.4 0.2 0

5.6

time

0

5.6 4

CR IP T

I and Z, mg C/L

S, mg C/L

1.5

x 10

1000 2000 3000 4000 5000

time

4

x 10

AN US

Figure 15. We show the population densities of our system for K = 3.9 and hS = 0.4. The focal host S, algae A, infected class I, and spores Z oscillate, while the competitor dies out.

3

2.5

M

AS

A

hS

1.5

SIZA

PT

1

ED

2

AC

CE

0.5

0

0

1

2

3

K

4

5

6

Figure 16. The bifurcation diagram for the SIZA system with respect to the focal host’s half-saturation constant hS and the algal carrying capacity K.

the incompetent host clearing the propagules and keeping their equilibrium density constant. As hS decreases, prevalence increases driven by a decrease in the density of the susceptible host and an increase in the densities of the infected and susceptible hosts and propagules. Namely, a more efficient host consumes more propagules and gets sicker. This in turn increases the production of propagules since the infected host always die releasing the propagules they contain back into the water column.

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In the region after the Hopf bifurcation occurs, we observed behavior such as period-doubling and torus bifurcations, and destruction of limit cycles. In the latter case, we start off with all five populations oscillating, and after the inferior competitor which is the incompetent host clears the infection, it gets out-competed by the focal host, which is the superior competitor. The system with the host and the algae keeps oscillating. This is due, we suspect, to an over-exploitation of the resource. In other instances, the competitor’s density crashes to zero, while there are endemic oscillations. In the variation of our system where the competitor is not present, we observed that for the same parameter region, the system does not oscillate. This implies that the competition destabilizes the system. A pairwise comparison of the equilibrium densities in the two systems, reveals that with the competitor present, the propagule density is always smaller, hence the competitor indeed clears them, and the infected class is always smaller too. On the other hand, the total host density (susceptible + infected host density) is (for most parameter values) smaller. The algal density is always smaller (set at the competitor’s minimal resource requirement), as is the disease prevalence. Hence, the competitor can be thought of as a diluter in the sense that it helps keeps the host healthier by removing the infective propagules, but because it is also a competitor, the total focal host population is smaller. Acknowledgements

M

We gratefully acknowledge support from the NSF through grants: DEB-1120804, DEB-1120316 and DUE-1129198. Appendix A. Equilibrium points: existence

ED

For the equilibrium ED , since D 6= 0, we obtain from Eq. (5), eD fD0 A∗ = dD + pD hD + A∗

PT

which in turn yields

CE

hD (dD + pD ) . eD fD0 − (dD + pD ) From Eq. (4) we obtain the value for D: A∗D =

r(1 −

fD (A∗ ) ∗ A∗ )= D K A∗

AC

which implies that ∗ DD =

rA∗ (K − A∗ ) reD A∗ (K − A∗ ) = . KfD (A∗ ) K(dD + pD )

For ES , since S 6= 0, we obtain from Eq. (1)

eS fS0 A∗ = d + pS hS + A∗

which in turn yields A∗S =

hS (d + pS ) . eS fS0 − d − pS

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From Eq. (4) 4 we obtain the value for S ∗ from r(1 −

fS (A∗ ) ∗ A∗ S )= K A∗

which implies that rA∗ (K − A∗ ) reS A∗ (K − A∗ ) = . KfS (A∗ ) K(d + pS )

CR IP T

SS∗ =

For ESIZA , Eq. (1-2) imply

AN US

For ESD , from Eq. (5), we obtain A∗ = A∗D . From Eq. (1) with S ∗ 6= 0 we obtain A∗ = A∗S . Hence, we need tho check the conditions for which AS = AD , namely hS (d + pS ) hD (dD + pD ) = . eS fS0 − d − pS eD fD0 − dD − pD If this true and the resulting value for A∗ is positive, then from Eq. (4) it follows that the equilibrium densities for S ∗ and D∗ satisfy A∗ rA∗ (1 − ) = fS (A∗ )S ∗ + fD (A∗ )D∗ . K

eS fS (A∗ ) − (d + pS ) I∗ = . ∗ S d + v + pI − ρeS fS (A∗ )

Then, Eq. (2) and the above equality yield

Next, Eq. (3 - 4) yield

(d + v + pI )A∗ eS fS (A∗ ) − (d + pS ) . µfS (A∗ ) d + v + pI − ρeS fS (A∗ )

M

Z∗ =

∗

PT

ED

λ + r(1 − AK ) (d + v + pI )A∗ eS fS (A∗ ) − (d + pS ) . σeS fS (A∗ )(d + v) µfS (A∗ ) d + v + pI − ρeS fS (A∗ ) From this, in turn we obtain the value for S ∗ as a function of A∗ : d + v + pI − ρeS fS (A∗ ) ∗ S∗ = I . eS fS (A∗ ) − (d + pS ) I∗ =

AC

CE

Finally, from Eq. (4) we can obtain the value of A∗ from the equation ∗   λ + r(1 − AK ) d + v + pI eS fS (A∗ ) − (d + pS ) A∗ )= 1 + . r(1 − K σeS fS (A∗ )(d + v) µ d + v + pI − ρeS fS (A∗ ) After calculations, this yields [r(d + v + pI )(v + pI − pS + eS fS0 (1 − ρ)) − rµσeS fS0 (d + v)((d + v + pI ) − ρeS fS0 )] (A∗ )3 + [rµσeS fS0 (d + v)[(d + v + pI − ρeS fS0 )K − (d + v + pI )hS ] + rhS (d + v + pI )(v + pI − pS ) − ((v + pI − pS ) + eS fS0 (1 − ρ))((λ + r)K − rhS )(d + v + pI)](A∗ )2 +

(d + v + pI )[rµσeS fS0 (d + v)KhS − ((λ + r)K − rhS )(v + pI − pS )hS − (λ + r)KhS [(v + pI − pS ) + eS fS0 (1 − ρ)]]A∗ − (d + v + pI )(λ + r)Kh2S (v + pI − pS ) = 0.

If we vary K and hS and keep the rest of the parameters at the default values, then it turns out that there are 2 positive real roots. Of the two positive values for A∗ only one results in positive values for the remaining four equilibrium densities. Hence, there is a unique ESIZA equilibrium in O.

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Finally, for ESIZAD , Eq. (5) yields A∗ = A∗D . Eq. (1-2) imply I∗ eS fS (A∗ ) − (d + pS ) = . S∗ d + v + pI − ρeS fS (A∗ ) Then, Eq. (2) yields (d + v + pI )A∗ eS fS (A∗ ) − (d + pS ) . µfS (A∗ ) d + v + pI − ρeS fS (A∗ )

Next, Eq. (3-4) yield I∗ =

∗

λ + r(1 − AK ) Z ∗. σeS fS (A∗ )(d + v)

Then, ∗

λ + r(1 − AK ) (d + v + pI )A∗ . σeS fS (A∗ )(d + v) µfS (A∗ )

AN US

S∗ =

CR IP T

Z∗ =

Lastly, Eq. (4) yields the value for D. D∗ =

rA∗ (1 −

A∗ K )

− fS (A∗ )(S ∗ + I ∗ ) . fD (A∗ )

Appendix B. Equilibrium points: stability

eS fS (A) − (d

ρeS fS (A)

−µ fSA(A) S

eS (S + ρI)fS0 (A)−  0 µ fSA(A) SZ

PT

CE AC

0



      0  fS (A) fS (A)  −(d + v + pI ) µ A S µ SZ 0 A     σeS fS (A)(d + v) −λ − fSA(A) (S + I) σeS fS0 (A)(d + v)I− − fDA(A) Z    0  fS (A) − fSA(A) Z − fDA(A) D (S + I)− . A   0  fD (A  D A    2A 0 −fS (A) 0 r(1 − K ) − fS (A)(S + I) −fD (A)    0 −fD (A)D    0 0 0 eD fD (A)D eD fD (A)−  (dD + pD )

ED

  +pS ) − µ fSA(A) Z      µ fSA(A) Z      − fSA(A) Z           −fS (A)       0

M

The Jacobian matrix for the dynamical system (1-5) is the following: 

For the trivial equilibrium E0 the Jacobian matrix is diagonal. The eigenvalues are −(d + pS ) < 0, −(d + v + pI ) < 0 −λ < 0, r > 0, −(dD + pD ) < 0. For the equilibrium EA , the eigenvalues are eS [fS (K)−fS (A∗S )], −(d+v+pI ) < 0, −λ < 0, −r < 0, and eD [fD (K) − fD (A∗D )]. If either K > A∗S or K > A∗D this equilibrium is linearly unstable. D∗ For the equilibrium ED the eigenvalues are −λ−fD (A∗D ) A∗D < 0, −(d+v +pI ) < D 0, eS [fS (A∗D ) − fS (A∗S )]. The sign of the last eigenvalue depends on the sign of

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∗ 0 b1 = eD DD fD (A∗D )fD (A∗D ) =

CR IP T

A∗D − A∗S , since the feeding curves are monotonically increasing. We also obtain two more that satisfy the quadratic polynomial 2A∗D 0 ∗ ∗ 0 Λ2 − [r(1 − ) − fD (A∗D )DD ]Λ + eD DD fD (A∗D )fD (A∗D ) = 0. K This can be written as Λ2 − a1 Λ + b1 = 0, where by the assumptions on the feeding curve, it follows ∗ 2 eD A∗D DD fD0 hD > 0. (hD + A∗D )3

For the other parameter value, we have A∗ K − hD − 2A∗D a1 = r D 0 fS (A∗S ) ∗ SS > 0 a2 = d + v + pI + λ + A∗S fS (A∗S ) ∗ f 2 (A∗ ) SS ) − µσeS (d + v) S ∗ S SS∗ , ∗ AS AS

it holds

p

a1 ±

AC

CE

a21 − 4b1 2 p −a ± a22 − 4b2 2 Λ2 + a2 Λ + b2 = 0 ⇔ Λ = . 2 From this, it follows that the eigenvalues are real negative and/or are complex conjugate with negative real part if and only if a1 < 0 and b2 > 0. It can be shown that rA∗S K − hS − 2A∗S a1 = . ∗ K(hS + AS ) hS + A∗S Λ2 − a1 Λ + b1 = 0 ⇔ Λ =

Hence we obtain two eigenvalues that are negative or have negative real part if K − hS − 2A∗S < 0, and from (15) it follows that b2 = (d + v + pI )(λ +

fS (A∗S ) ∗ SS )(1 − R0 ). A∗S

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From the above conditions it follows that the disease-free equilibrium is linearly asymptotically stable when A∗S < A∗D , K − hS − 2A∗S < 0, and R0 < 1.

CR IP T

For the ESIZA equilibrium, the characteristic polynomial contains the factor eD fD (A∗ ) − (dD + pD ) = eD (fD (A∗ ) − fD (A∗D )),

so since fD is a monotonically increasing function, it follows that there is a positive eigenvalue if A∗SIZA > A∗D . References

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AC

CE

´ ceres, S. Duple, and P. Lee, School of Integrative Biology, University C. E. Ca of Illinois at Urbana–Champaign, Urbana, IL 61801, G. Davis, A. Koss, and Z. Rapti, Department of Mathematics, University of Illinois at Urbana–Champaign, Urbana, IL 61801, S. R. Hall, Department of Biology, Indiana University, Bloomington, IN 47405