System analysis: Competition and coexistence

Intro

• Competition is common in all kinds of (eco)systems.

• Competition requires limited essential resources like nutrients, light, or space.

• Species coexistence and thus biodiversity are necessarily linked to competition.

• Models can gain insights into the nature of competition and the conditions for coexistence.

• Compared to experiments, we can perfectly disentangle / control all involved mechanisms.

About the example

• We consider a pair of two planktonic algae species in shallow lake (i.e. a CFSTR).

• We account for competition for nutrients (P, N) but ignore light for now.

• We do not model common causes of mortality but consider flushing (outflow) as the sole loss process.

Load the model

rm(list=ls())                                  # initial clean-up
library("rodeo")                               # load package

workbook <- "models/coexistence.xlsx"          # adjust path as necessary
m <- buildFromWorkbook(workbook)               # build model

Function implementation

• N2Pin represent the N:P ratio in the reactor’s input (using mass units).

• Implemented as a periodic function with amplitude max - min and cycle duration given by interval.

• Choosing min == max results in a constant value.

N2Pin <- function(time, mini, maxi, interval) {
  mini + 0.5*(sin(time/interval * 2 * 3.1415) + 1) * (maxi - mini)
}

Function implementation

redf <- 16*14 / (1*31)   # Redfield ratio of 16:1 in mass units
times <- 0:96
plot(times, N2Pin(times, mini=redf/5, maxi=redf*2, interval=24),
  type="l", xlab="time", ylab="N:P mass ratio")

Case 1: Identical competitors

• Run the model with default parameters.

• By the defaults, species A and B are identical.

• N:P in inflow is constant.

x <- m$scenarios(times=0:100)

Case 1: Identical competitors

• Species A and B coexist in the reactor.

• This is true for arbitrary non-zero initial values.

• No surprise, because A and B are indistinguishable.

Case 2: Superiority of a player

• We grant an advantage to species A with regard to one resource (here: Phosphorus).

• The advantage is reflected by the lower half-saturation constant (hP_A < hP_B).

• We let A and B compete under two scenarios: P limitation and N limitation.

common <- c(hP_A=0.005, hP_B=0.01)                # hP_A < hP_B
sc <- list(
  P.limitation=c(common, minN2P=10, maxN2P=10),   # excess of N 
  N.limitation=c(common, minN2P=3, maxN2P=3)      # excess of P
)
x <- m$scenarios(times=0:100, scenarios=sc)

Case 2: Superiority of a player

• If P is the limiting resource, the superior competitor (species A) dominates the system while species B goes extinct. The winner takes it all.

• If N is the limiting resource, we still experience coexistence (because hN_A = hN_B).

• Hence, superiority regarding the uptake of a resource only pays out if that resource is actually limiting. No advantage from unnecessary skills.

Case 3: Contrasting advantages

• In classical theory, coexistence requires niches (i.e. each species is adapted to the specific conditions in a sub-habitat or capable of exploiting an exclusive resource).

• But here, species A and B exploit identical resources (P, N) and in a homogeneous environment.

• However, if suitable habitats for two distinct species do not exist in the spatial domain they can still exist in the time domain.

Case 3: Contrasting advantages

• We equip A and B with contrasting advantages (hP_A < hP_B but hN_A > hN_B).

• Depending on N and P supply, either A or B is the better competitor.

• We study scenarios of nutrient supply, including a periodical switch between P and N limitation.

common <- c(tau=5, hP_A=0.01, hP_B=0.02, hN_A=0.2, hN_B=0.1)
sc <- list(
  low.N2P=  c(common, minN2P=5,   maxN2P=5),
  high.N2P= c(common, minN2P=9.5, maxN2P=9.5),
  var.N2P=  c(common, minN2P=5,   maxN2P=9.5, intN2P=20)
)
x <- m$scenarios(times=0:300, scenarios=sc)

Case 3: Contrasting advantages

• In a constant environment, there is no coexistence.

• Temporal variability can lead to coexistence.

• Amplitude and frequency must be in a suitable range to serve both competitors with a sufficient niche.

• The maintenance of species diversity through temporal variation has many facets.

• You may want to read about the intermediate disturbance hypothesis (IDH). In that context “disturbance” is used in a broad sense and includes, e.g., the reset of successions.

Case 3: State-space view

Let’s draw the trajectories in state-space for the scenario with periodically varying N:P supply.

# simulate particular scenario again
x <- m$scenarios(times=0:300, scenarios=sc["var.N2P"], plot.vars=F)

# plot trajectories with color indicating time
plot(B ~ A, data=x, type="l", lty=3)
intens <- seq(1, 0, length.out=nrow(x))
points(B ~ A, data=x, pch=1, col=rgb(intens, 0, rev(intens)))
legend("top", bty="n", pch=c(1,1), col=c("red","blue"),
  legend=c("Initial state","Late states"))

Mechanisms behind diversity (1)

Mechanisms behind diversity (2)

Mechanisms behind diversity (3)

Mechanisms behind diversity (4)

• Consider this review paper to learn more about the mechanisms of (bacterial) competition.

• In real-word ecosystems, it is often difficult to disentangle the reasons of apparent coexistence.

• In aquatic systems, you will often find a combination of both co-existence (niches) and mere co-presence (import from external habitats through hydrological connections).