Ecological rescue under environmental change Supplementary Material 1: Description of full metacommunity model *Pradeep Pillai 1, Tarik C. Gouhier 1, and Steven V. Vollmer 1 * Corresponding author: pradeep.research@gmail.com 1 Marine Science Center Northeastern University 430 Nahant Road Nahant, MA 01908 1
Within-host equilibrium densities of microbes Within host patch communities, the local equilibrium densities of microbial species z and y, when they co-occur in a patch, are determined using a simple Lotka-Volterra resource competition model as described in MacArthur and Levins (1967). The competitive interactions within a single host determine what type of communities will be observed at the regional scale. For example, with four microbial species (x, z, y 1, y ), and noting that x and z cannot coexist together, there are eleven possible microbial communities within Environment 1, excluding the empty host community with no microbes, i.e., {x}, {z}, {y 1 }, {y }, {y 1, x}, {x, y }, {y 1, z}, {z, y }, {y 1, x, y }, {y 1, z, y }, and {y 1, y }. Within Environment there are four additional microbial communities: {x}, {y 1, x}, {x, y }, and {y 1, x, y }. We use the lower case expressions ˆx, ẑ and ŷ to represent the equilibrium population densities of mutualists, strong cheaters and weak cheaters in a local community, respectively. We will assume that all species have the same carrying capacity K = 1, such that any single species communities composed of either x, z or y, will all be of the same size. For modelling the outcome of local species competition in communities with more than one species we utilize the approach of MacArthur and Levins (1967), and define each species niche by a normal distribution along a single linear niche gradient with a standard deviation of σ = 1. We let α represent the niche overlap between a weak cheater and either a strong cheater or a mutualist. Thus α in our model represents the competition coefficient between a weak cheater and strong cheater, and can be calculated (in the case of the competitive effects of the weak cheater on the strong cheater) as follows: α zy = ˆ ˆ ẑ(r)ŷ(r)dr, (1) ẑ(r) dr where r is the resource.
In the case of a two species community composed of a strong cheater and a weak cheater, the respective densities of each would be as follows: ẑ = K α zy ŷ, and ŷ = K α zy ẑ, which gives us ẑ = ŷ = K 1+α zy. Since mutualists are a competitively subdominant microbe, a mixed community of mutualists and weak cheaters will result in the mutualist achieving an equilibrium density determined by the amount of resource remaining after the weak cheater has achieved carrying capacity; i.e., ˆx = (1 α xy )K and ŷ = K. Recall that since mutualists and strong cheaters occupy identical niches, they cannot coexist in the same community, and mutualists will always be excluded in the presence of strong cheaters. In the case where there are two weak cheaters (y 1 and y ) that are symmetrically arranged around, and equally distant from, the mean of the strong cheater/mutualist niche, we have an additional possible two species community, {ŷ 1, ŷ }, as well as two possible three species communities: {ŷ 1, ẑ, ŷ } and {ŷ 1, ˆx, ŷ }. If α yy is the niche overlap between y 1 and y, then the densities of each species in the two weak cheater community is similar to the two species community described above: ŷ 1 = ŷ = K 1+α yy. For the three species communities we can derive the following expressions for α using (1): ( ) d α zy = exp, () 4σ ( ) d α yy = exp, (3) σ for standard deviation σ, and niche distance between mutualist and weak cheater d. The expressions for the equilibrium densities in the three species community {ŷ 1, ẑ, ŷ } are 3
ẑ = K (1 + α yy) α zy K, (4) 1 + α yy αzy ŷ 1 = ŷ = (1 α zy) K. (5) 1 + α yy αzy Note that for all three species to be present (i.e., have density > 0) the following condition must hold: 1 > α. which gives a threshold condition of α < 0.544. If the condition 1+α 4 zy does not hold then the strong cheater cannot coexist with the two weak cheaters within a patch at the same time due to insufficient niche space between y 1 and y to allow z to invade. In the {ŷ 1, ˆx, ŷ } community the equilibrium expression for the two weak cheaters is, as before, ŷ 1 = ŷ = ˆx = (1 α zy )K. K 1+α yy, while the equilibrium expression for the mutualist becomes Metacommunity Model The metacommunity model involving weak cheaters was implemented as a system of thirteen ordinary differential equations (ODEs). The dynamics of the ODE system were simulated in MATLAB using solvers ode45 and ode3s for periods ranging between 10,000 and 50,000 time steps, which were sufficient to ensure equilibrium dynamics. Each equation in the metacommunity model represents the metapopulation density of hosts in a particular environment with a particular type of community. The metapopulation densities give the fraction of total landscape that is occupied by a host with a given microbial community; as such, density values are between 0 and 1. The equations are shown below. For our differential equations system we use state variables H 1 and H to track the density of hosts in Environments 1 and. Furthermore, we use sub-subscripts to indicate 4
the type of microbial community in the host. For example, H 1X, H 1Z, and H 1Y1 represent the density of hosts in Environment 1 that are occupied exclusively by mutualist x, strong cheater z, and weak cheater y 1, respectively. Recall that because the niches of y 1 and y are arranged symmetrically around, and are equidistant from the resource niche of the mutualist and strong cheater, the effects of both y 1 and y are identical and indistinguishable when in separate hosts. As a result, we can subsume for computational purposes both y 1 and y into a single generic weak cheater category and variable, y. This significantly reduces the complexity and dimensionality of our equation system because we can now assume the state variables representing hosts occupied by y 1 and y, e.g., H 1Y1 and H 1Y, are equivalent and can be combined into a single state variable, in this case H 1Y. Thus, by assuming H 1Y1 = H 1Y, we can claim that the total density of hosts occupied by either y 1 or y is simply 1 H 1 Y = H 1Y1 = H 1Y. Similarly, assuming that H 1XY1 = H 1XY, H 1ZY1 = H 1ZY, and H XY1 = H XY allows us to claim that 1 H 1 XY = H 1XY1 = H 1XY, 1 H 1 ZY = H 1ZY1 = H 1ZY, and that 1 H XY = H XY1 = H XY. In the system of differential equations we use H 1, H 1X, H 1Z, H 1Y, and H X as state variables to represent, respectively, the total metapopulation density of hosts in Environment 1, the density of mutualist-occupied hosts in Environment 1, the density of strong cheateroccupied hosts in Environment 1, the density of a single weak cheater-occupied hosts in Environment 1, and the host density in Environment when occupied only by a single mutualist population. There are seven additional state variables to represent host densities for those cases where weak cheater microbes y 1 and y coexist with other microbial species in a single host: H 1XY and H 1XY Y, for when hosts are occupied by mutualists coexisting with either one or both weak cheater microbes within Environment 1; H 1ZY and H 1ZY Y for hosts occupied by strong cheaters with one or both weak cheater microbes within Environment 1; H 1Y Y for hosts occupied by both weak cheater microbes within Environment 1; and H Y and H Y Y for mutualist-occupied hosts within in Environment with one or both weak cheater species. New host patches are created by colonisation of empty space at a rate c, while 5
the loss rate of existing hosts in Environment 1, and in Environment with mutualists, is given respectively, by e 1 and e. As well, new local microbial populations within hosts are created by colonisation of existing hosts with available resources at a rate of β, while local microbial populations go extinct at rate δ. For a given microbial species, both β and δ are scaled according to their local within-host population density. Thus, microbial colonization and extinction rates for a given species are represented by subscripts which indicate the species, and sub-subscripts which indicate the microbial community or species group that the reference species coexists with within a host. For example, β XY is the colonization rate of mutualists from a host that is also occupied by a single weak cheater species. Similarly, δ ZY Y is the local extinction rate of a strong cheater population in the presence of both weak cheater species in a single host. The system of differential equations for the host-microbiome metacommunity is as follows: dh 1 = c(h 1 + H X + H Y + H Y Y )(E 1 H 1 ) e 1 H 1, (6) dh 1X = β X H X (H 1 H 1X H 1Z H 1Y H 1XY H 1ZY H 1Y Y H 1XY Y H 1ZY Y ) (e 1 + δ X + β Y H Y + β Z H 1Z ) H 1X + δ YX H 1XY, (7) dh 1Z = β Z H 1Z (H 1 H 1Z H 1Y H 1XY H 1ZY H 1Y Y H 1XY Y H 1ZY Y ) (e 1 + δ Z + β Y H Y ) H 1Z + δ YZ H 1ZY, (8) 6
dh 1Y = β Y H Y (H 1 H 1X H 1Z H 1Y H 1XY H 1ZY H 1Y Y H 1XY Y H 1ZY Y ) (e 1 + δ Y + β X X + β Z H 1Z + 0.5 β Y H Y ) H 1Y + δ XY H 1XY + δ ZY H 1ZY + δ YY H 1Y Y, (9) dh 1XY = β Y H 1Y H 1X + β X H X H 1Y (e 1 + δ XY + δ YX ) H 1XY β Z H 1Z H 1XY 0.5 β Y H Y H 1XY + δ YY X H 1XY Y, (10) dh 1ZY = β Y H Y H 1Z + β Z H 1Z (H 1Y + H 1XY ) (e 1 + δ ZY + δ YZ ) H 1ZY 0.5 β Y H Y H 1ZY + δ YY Z H 1ZY Y, (11) dh 1Y Y = 0.5 β Y H Y H 1Y (e 1 + δ YY )H 1Y Y β X X H 1Y Y β Z Z H 1Y Y + δ XY Y H 1XY Y + δ ZY Y H 1ZY Y, (1) dh 1XY Y = 0.5 β Y H Y H 1XY + β X H X H 1Y Y e 1 H 1XY Y (δ YY X + δ XY Y )H 1XY Y β Z H 1Z H 1XY Y, (13) 7
dh 1ZY Y = 0.5 β Y H Y H 1ZY + β Z H 1Z H 1Y Y e 1 H 1ZY Y (δ YY Z + δ ZY Y )H 1ZY Y + β Z H 1Z H 1XY Y, (14) dh = c(h 1 + H X + H XY + H XY Y )(E H ) µ(h H X H XY H XY Y ) (15) (e + δ X )H X β Z H 1Z (H X + H XY + H XY Y ) δ XY H XY δ XY Y H XY Y, dh X = β X H X (H H X H XY H XY Y ) (e + δ X )H X (16) β Y H Y H X β Z H 1Z H X + δ YX H XY dh XY = β Y H Y H X (e Y + δ XY )H XY δ YX H XY β Z H 1Z H XY (17) 0.5 β Y H Y H XY + δ YY X H XY Y, dh XY Y = 0.5 β Y H Y H XY e Y Y H XY Y (δ YY + δ XY Y )H XY Y β Z H 1Z H XY Y, (18) where β X H X = (β X H 1X + β XY H 1XY + β XY Y H 1XY Y + β X H X + β XY H Y + β XY Y H Y Y ), β Z H 1Z = (β Z H 1Z + β ZY H 1ZY + β ZY Y H 1ZY Y ), and β Y H Y = (β Y H 1Y + β YZ H 1ZY + β YX H 1XY + β Y Y H 1Y Y + β YY X H 1XY Y + β YY Z H 1ZY Y + β YX H XY + β YY X H XY Y ). 8
E 1 and E give the relative proportions of habitats Environment 1 and Environment. The value of e Y is scaled according to the density of the local mutualist population in a mixed mutualist/weak cheater community. In this model, the scaling of e Y with local population density is defined by the expression e Y = e (ˆxy /ˆx) u, where u is the scaling exponent. e Y Y is defined in a similar manner: e Y Y = e (ˆxyy /ˆx) u. As mentioned, the extinction rates of local microbial populations also depend on local population density. In mixed communities with multiple species, the baseline local extinction rate of any given species, δ, will be scaled according the ratio of the equilibrium density of the species in a mixed community relative to the equilibrium density of the species on its own. For example, in the case of the mutualist co-occurring with a single weak cheater we have δ XY = δ (ˆxy /ˆx) v, with scaling exponent v. The same holds for other mixed microbial communities, such as δ ZY and δ YZ. For results shown, u = 0.5 and ν = 1. Finally, the rate of colonization of new host patches by microbes β is determined by the rate of outgoing migrants from a host, and thus is linearly scaled by local population densities in mixed community patches, as is shown in the example of β XY : β XY = β ˆx y. Dynamics of the ODE systems were run with random initial densities of mutualist and weak cheaters above a threshold required to allow the mutualist and host to achieve a positive equilibrium after several thousand time steps. Strong cheaters were then introduced into the system as an invasive at low host-occupancies (Z 0.005), after which dynamics were run between 10 000 to 50 000 time steps. 9