Trait diversity promotes stability of community dynamics

Abstract

The theoretical exploration of how diversity influences stability has traditionally been approached by species-centric methods. Here we offer an alternative approach to the diversity–stability problem by examining the stability and dynamics of size and trait distributions of individuals. The analysis is performed by comparing the properties of two size spectrum models. The first model considers all individuals as belonging to the same “average” species, i.e., without a description of diversity. The second model introduces diversity by further considering individuals by a trait, here asymptotic body size. The dynamic properties of the models are described by a stability analysis of equilibrium solutions and by the non-equilibrium dynamics. We find that the introduction of trait diversity expands the set of parameters for which the equilibrium is stable and, if the community is unstable, makes the oscillations smaller, slower, and more regular. The stabilizing mechanism is the variation in growth rate between individuals with the same body size but different trait values.

Appendices

Appendix 1: Boundary condition

We derive the lower boundary condition N(m _b,M) in the trait-based model on the basis of equilibrium size spectrum (ESS) theory (Andersen and Beyer 2006). Moreover, we assume that the total offspring recruitment in the trait-based model equals the recruitment in the community model, that is,

$$ \int_{M_0}^{M_{\rm b}} N(m_{\rm b},M){\rm d}M = \kappa m_{\rm b}^{-\uplambda}. $$

(11)

The ESS theory involves two assumptions. On one hand, the community size spectrum is of a power law state, that is, $\overline{N}_{\rm c}(m)=\kappa m^{-\uplambda}$. On the other hand, the feeding level, indicating the mount of energy routed to individual somatic growth, is constant. Under these two conditions, Andersen and Beyer (2006) found the analytical solution to Eq. 1

$$ \overline{N}(m,M) = k_0 M^{2n-q-3+a}m^{-n-a}\left(1-\left(\frac{m}{M}\right)^{r-n}\right)^{\frac{a}{r-n}-1}, $$

(12)

where a = β ^{2n − q − 1}/α is the physiological predation rate and k ₀ a constant satisfying $\int_m^{\infty} \overline{N}(m,M){\rm d}M = \kappa m^{-\uplambda}$. The exponent of community size spectrum is found to be $\uplambda = 2+q-n$. Due to Eq. 11, the lower boundary condition is set as

$$ N(m_{\rm b}, M) = f_0 \overline{N}(m_{\rm b}, M)/k_0 \simeq f_0 M^{2n-q-3+a}m_{\rm b}^{-n-a}. $$

(13)

Therefore, f ₀ is calculated by evaluating

$$ f_0 = k_0 \kappa m_{\rm b}^{-\uplambda+n+a}/\int_{M_0}^{M_{\rm b}}\overline{N}(m_{\rm b}, M){\rm d}M .$$

(14)

Since resource spectrum is fixed, encountered food of offspring can be calculated analytically

$$\begin{array}{rll} E(m_{\rm b}) & = & \gamma m_{\rm b}^{q} \int_{10^{-6}}^{10^{-3}}\kappa w^{1-\uplambda}\varphi(m_{\rm b}/w){\rm d}w \\ &\simeq& \gamma m_{\rm b}^{q} \int_{0}^{\infty}\kappa w^{1-\uplambda}\varphi(m_{\rm b}/w){\rm d}w \\ &=& \gamma\kappa\beta^{\uplambda-2}e^{(\uplambda-2)^2\sigma^2/2}m_{\rm b}^{2+q-\uplambda}. \end{array}$$

Note that $\uplambda =2$ when q = n. Thus, E(m _b) is independent of the width of selection function σ and the preferred predator–prey mass ratio β.

Appendix 2: Averaging growth efficiency

The average of growth efficiency is calculated based on the ESS theory, and our calculation is in principle a replicate of Andersen and Beyer (2006) and Andersen et al. (2009). In order for completeness, we present it here briefly.

The ESS theory assumes an infinite size range for community size spectrum, i.e., $\overline{N}_{\rm c}(m) = {km}^{-\uplambda}, \ 0< m < \infty$. The advantage is that we can get a closed form when doing integration to obtain individual encountered food (Eq. 15) and mortality (Eq. 18), whereas in our trait-based model and community model, size spectrum is, however, truncated for numerical calculations. The individual encountered food can be calculated as

$$ E(m) = \gamma m^q \int_{0}^{\infty} w\overline{N}_{\rm c}(w)\varphi(m/w){\rm d}w = \kappa \alpha_e m^n, $$

(15)

where $\alpha_e =\gamma \beta^{\uplambda-2}e^{(\uplambda-2)^2\sigma^2/2}$. Thus, individual somatic growth rate is

$$ g(m,M) = \alpha \left(1-\left(\frac{m}{M}\right)^{r-n}\right)E(m). $$

(16)

According to Eqs. 12 and 16, the average growth rate is calculated as

$$ g_{\rm c}(m) = \frac{1}{\overline{N}_{\rm c}}\int_m^{\infty} g(m,M)\overline{N}(m,M){\rm d}M = \alpha_{\rm c} m^n , $$

(17)

where the coefficient α _c is to be determined.

The mortality rate is

$$ \mu(m) = \int_0^{\infty} \gamma w^q \overline{N}_{\rm c}(m) \varphi(w/m){\rm d}w = \alpha_p m^{1+q-\uplambda}, $$

(18)

where $\alpha_p = \gamma \kappa e^{(1+q-\uplambda)^2\sigma^2/2}\beta^{1+q-\uplambda}$. Inserting the average growth rate (Eqs. 16 and 18) into the stationary equation of Eq. 10, we obtain

$$ \frac{\partial }{\partial m}(N_{\rm c}(m) g_{\rm c}(m)) = \mu(m)N_{\rm c}(m). $$

(19)

By solving the equation (Eq. 19), the exponent of the community size spectrum is found to be $\uplambda = n + \alpha_p/\alpha_{\rm c}$. Notice that $\uplambda = 2+q-n$, and then α _c = α _p / (2 + q − 2n).

Dividing Eqs. 17 by 15, the average growth efficiency is

$$ \overline{\varepsilon}_I = \frac{\beta^{2n-q-1}}{2+q-n} e^{((n-1)^2-(q-n)^2)\sigma^2/2} . $$

(20)

Appendix 3: Tracing equilibrium and determining stability

To solve the non-equilibrium dynamics to the trait-based size-spectrum model, we refer readers to Hartvig et al. (2011) where detailed numerical program is included and the key point is how to discretize the McKendrick–von Foerster equation which is presented in Eq. 21. Analogously, the dynamics of the community size spectrum can also be solved. In the following, the focus is how to continuously trace equilibrium solution as a function of a free parameter (e.g., the diet width σ). To this end, we apply Newton’s parameter continuation (Kuznetsov 1994) combined with the semi-implicit upwind scheme (Hartvig et al. 2011). Since body size spans several orders of magnitude, logarithmical size is preferred. Details of the logarithmical transformation can be referred to Benoît and Rochet (2004). We describe our technical approach by taking the community model as an example.

Discretization

Set x = log(m) and discretize x as x _i  = log(m _b) + Δx, i = 0, ⋯ , k, where x _k  < log(M _b) < x _k + 1. In addition, let $u_i^t = u(x_i, t)\Delta x = N(m(x_i))m(x_i)\Delta x$ represent the total number of individuals in the range of [x _i − 1, x _i ]. Then the discretization of the McKendrick–von Foerster Eq. 10 can be described as follows:

$$ \frac{u_i^{t+\Delta t}-u_i^{t}}{\Delta t} = -\frac{ g_i^t u_i^{t+\Delta t} -g_{i-1}^t u_{i-1}^{t+\Delta t} }{\Delta x} - \mu_i^t u_i^{t+\Delta t}, $$

(21)

where

$$ g_i^t = \overline{\varepsilon}_I\gamma e^{(q-1)x_i}\sum\limits_j e^{x_j} u_j^t \varphi(x_i-x_j), $$

(22)

and

$$ \mu_i^t = \gamma \sum\limits_j e^{qx_j}u_j^t\varphi(x_j-x_i). $$

(23)

Set $U^t = (u_1^t, \cdots, u_k^t)^{\prime}$, $B = (\frac{\Delta t}{\Delta x}g_0 u_0, 0, \cdots, 0)^{\prime}$, where the subscript indicates the transpose of a matrix or a vector. g ₀ is completely determined by the resource spectrum and thus constant.

Equation 21 has an equivalent form

$$ A(U^t)U^{t+\Delta t} - U^t -B = 0, $$

(24)

where A(U) is a do-bidiagonal matrix and along the main and lower diagonals are vectors a = (a _i ) and b = (b _i ), respectively. Moreover, the entries in these two vectors are

$$ a_i = 1+ \frac{\Delta t}{\Delta x}g_i^t + \Delta t \mu_i^t $$

(25)

and

$$ b_i = -\frac{\Delta t}{\Delta x}g_i^t. $$

(26)

SetF(U): = A(U ^t)U ^t + Δt − U ^t − B. Making an explicit dependence of F on the free parameter leads to

$$ F(U, \sigma)= 0. $$

(27)

Newton’s continuation

Now it is ready to perform the Newton’s continuation using Eq. 27. Detailed description is referred to Kuznetsov (1994). We here present how to calculate the Jacobian matrix of Eq. 27 for given equilibrium U and σ.

Define dFdU to be the Jacobian matrix, which is of k ² dimension, and set c _j to be a row vector of dimension k whose entries are all zero except for the j-th component which is exactly 1. The dFdU can be calculated as follows: Noticing the special structure of the matrix A(U), the first row of the Jacobian is

$$ {\rm d}F{\rm d}U(1,:) = \left(\frac{\Delta t}{\Delta x}\frac{{\rm d}g_1}{{\rm d}U} + \Delta t \frac{{\rm d}\mu_1}{{\rm d}U} \right)u_1 + (a_1-1)c_1, $$

(28)

and for j > 1

$$ \begin{array}{rll} {\rm d}F{\rm d}U(j,:) &=& -\frac{\Delta t}{\Delta x} \frac{{\rm d}g_{j-1}}{{\rm d}U}u_{j-1} + b_{j-1}c_{j-1} \\ & & + \left(\frac{\Delta t}{\Delta x}\frac{{\rm d}g_j}{{\rm d}U} + \Delta t \frac{{\rm d}\mu_j}{{\rm d}U} \right)u_j+ (a_j-1)c_j, \end{array}$$

(29)

where dg _j /dU and dμ _j /dU are row vectors with the following form:

$$ \frac{{\rm d}g_j}{{\rm d}U} = \overline{\varepsilon}_I \gamma e^{({(q-1)}x_j)} [e^{x_1}\varphi(x_j-x_1), \cdots, e^{x_k}\varphi(x_j-x_k) ], $$

(30)

$$ \frac{{\rm d}\mu_j}{{\rm d}U} = \gamma [e^{qx_1}\varphi(x_1-x_j), \cdots, e^{qx_k}\varphi(x_k-x_j) ]. $$

(31)

To calculate the derivative of F with respect to the free parameter, we use the forward finite difference as an approximation, i.e.,

$$ {\rm d}F{\rm d}\sigma = (F(U, \sigma+ \delta) -F(U, \sigma))/ \delta, $$

(32)

where δ =  max {σδ ₁, δ ₂}.

Implementation of the parameter continuation in the trait-based model is similar but more challenging. Apart from the discretization along the direction of individual body size, the trait is also discretized evenly on the logarithmical scale. Thus, the matrix in Eq. 24 is now a black don-bidiagonal of K ² dimension with $K=\sum_{i=1}^{L}k_i$, where k _i is the number of discretized points for species i and L is the number of discretized species. Each block matrix has the same structure as the matrix in Eq. 24 with entries Eqs. 25 and 26.

After some initial experimentation, we found that setting $\Delta x =0.1, \Delta t= 0.02, \delta_1 = 10^{-4}, \delta_1 = 10^{-7}, L = 20$ could produce reliable result without causing too much computational effort, even though the precise choice of discretization affects the resolution of the approximation.

Determining the stability of equilibrium

In analogy to Eq. 21, discretization of the McKendric–von Foerster equation at given equilibrium U gives rise to

$$ \frac{\partial }{\partial t} u_i(t) = - \frac{g_iu_i-g_{i-1}u_{i-1}}{\Delta x} -\mu_iu_i, $$

(33)

where g _i and μ _i are similar with Eqs. 22 and 23. Rewriting the right-hand side of Eq. 33 as a matrix leads to

$$ G(U) = A_{\rm s}(U) +B_{\rm s}, $$

(34)

where A _s and B _s are similar to A and B but the entries are $-\frac{a_i-1}{\Delta t}$ and $-\frac{b_i}{\Delta t}$ in A _s and B/Δt in B _s. The Jacobian matrix of G(U) with respect to U can be calculated in analogy to Eqs. 28 and 29. Stability of equilibrium U is determined through the maximum real part of the Jacobian matrix of G(U) with positive value meaning unstable and vice versa.