Theory of invasion extinction dynamics in minimal food webs

When food webs are exposed to species invasion, secondary extinction cascades may be set off. Although much work has gone into characterizing the structure of food webs, systematic predictions on their evolutionary dynamics are still scarce. Here we present a theoretical framework that predicts extinctions in terms of an alternating sequence of two basic processes: resource depletion by or competitive exclusion between consumers. We first propose a conceptual invasion extinction model (IEM) involving random fitness coefficients. We bolster this IEM by an analytical, recursive procedure for calculating idealized extinction cascades after any species addition and simulate the long-time evolution. Our procedure describes minimal food webs where each species interacts with only a single resource through the generalized Lotka-Volterra equations. For such food webs ex- tinction cascades are determined uniquely and the system always relaxes to a stable steady state. The dynamics and scale invariant species life time resemble the behavior of the IEM, and correctly predict an upper limit for trophic levels as observed in the field.

Figure 1

Invasion extinction model. The schematic exemplifies the consequence of adding a species ${\mathcal{S}}_{\mathrm{new}}$ with decay rate $r_{\mathrm{new}}$ at the basal level (blue circles). The blue shaded cube denotes the basic nutrient source, e.g., sugar. When adding ${\mathcal{S}}_{\mathrm{new}}$, all consumer-resource species pairs (green and blue circles) corresponding to basal-level decay rates $r_i>r_{\mathrm{new}}$ become eliminated; in the example these are the species ${\mathcal{N}}_3$ and ${\mathcal{N}}_4$, corresponding to the the decay coefficients $r_3$ and $r_4$, respectively. Subsequently, a new level-two species may be added to ${\mathcal{N}}_{\mathrm{new}}$. This dynamics mimics species replacement as it would be seen in a dynamical implementation of the “kill the winner” scenario for microbial ecosystems [16].

Figure 2

Definitions. (a) Simple food web consisting of four trophic levels. We define the nutrient source to be trophic level zero. The trophic level of any species is the average trophic level of its resources plus 1. Colored circles mark species, light blue ovals mark pairings of species, and symbol (T) marks the top species, i.e.. species without consumers. Side branches are marked by the symbol “SB.” A branching species (or branching point) is marked by “B.” $N_c^{*}=r^{\prime \prime }/\beta$. (b) An example of a pairing, consisting of a “free” consumer and a “controlled” resource species. (c) Definition of $r_{\mathrm{eff}}$ for a controlled species: $r_{\mathrm{eff}}$ equals the sum of the decay coefficient of the species itself and the steady-state density of its controlled consumer(s). (d) Illustration of the condition for feasibility at branching points, showing the comparison between ${\tilde{r}}_{\mathrm{eff},f}^{\left(l\right)}$ and $r_{\mathrm{eff},c}^{\left(l\right)}$ for two branches formed by a free, respectively controlled, species. (Figure reproduced from the literature [14].)

Figure 3

Rules for invasions and species extinctions. (a) Introduction of new species (labeled “7”) eliminates species 6 and 4 by competitive exclusion and species 5 by resource depletion (since $r_7<r_6$ and $r_7<r_4$). The node area in the left and right schematic is proportional to species concentrations; in the central schematic, the node area is proportional to the species' decay coefficients $r_i$. Colors reflect the time of species addition (blue is historic, red is recent). (b) Simulated population dynamics with extinction of species 4, 5, and 6. (c) Invasion rules emphasizing the iterative elimination process: (I) invasion by a new species, (II) competitive exclusion, and (III) feasibility. The definitions of decay rates in (c) are as follows: $r_{\mathrm{eff}}^{\left(l\right)}\equiv \left[r^{\left(l\right)}+{\sum }_c{N}_c^{*(l+1)}\right]/{\beta }_i^{\left(l\right)}$, where the controlled species concentrations $N_c^{*(l+1)}$ are determined from Eq. (6); $r_{\mathrm{coll},c}^{\left(l\right)}\equiv r_{\mathrm{eff},c}^{\left(l\right)}+r_{\mathrm{coll},f}^{(l+1)}/{\beta }^{\left(l\right)}$; $r_{\mathrm{coll},f}^{\left(l\right)}\equiv max(r_{\mathrm{coll},c,1}^{(l+1)},\cdots ,r_{\mathrm{coll},c,n}^{(l+1)})$.

Figure 4

Evolution of fitness and diversity. (a) Decay rate $\mathrm{\Delta }r\equiv r-r_{\min }$ for all species vs. time in units of invasion attempts using $r_{\min }=0.02$. Blue, green, orange, and red points mark the presence of a species at trophic levels 1, 2, 3, and 4. The blue line marks the lowest decay coefficient at the basal level. The gray line marks the median decay rate; note its increase whenever a new superior bottom species appears. Note the double-logarithmic scales. (b) Total species richness vs. time.

Figure 5

Species origination and extinction. (a) Time evolution for ${10}^6$ attempted species additions. Origination and existence time shown on horizontal and vertical axes, respectively. Inset: Origination-extinction matrix from fossil record data [28]; matrix reproduced from the literature [27, 29]. (b) Magnification of the first ${10}^4$ addition attempts, i.e., the region within the red box in the lower left corner of (a). Inset: Cumulative species lifetime distribution for each trophic level (blue to red for levels 1 to 4) and combined (black curve). the straight line indicates a power law with exponent $-2$. (c) Average distribution of trophic levels for the data in Figs. 4 and 4. (d) Trophic levels obtained for the free-living species of seven high-resolution aquatic food web data sets [15].