on the spatial scales of population fluctuations of two competing - - PowerPoint PPT Presentation
Effects of harvesting and strength of competition on the spatial scales of population fluctuations of two competing species Javier Jarillo Daz Universidad Complutense de Madrid (Spain) Mathematical Perspectives in Biology ICMAT
Javier Jarillo Díaz Universidad Complutense de Madrid (Spain) Mathematical Perspectives in Biology − ICMAT − February 2016
One species models. Effects of:
Environmental fluctuations (Moran effect). Dispersal.
Two species models. Effects of:
Harvesting. Competition.
Conclusions.
Spatial population synchrony: correlation of temporal fluctuations in
population size between neighbor localities.
Causes:
Environmental factors (e.g., temperature and humidity). Dispersal. Inter-specific interactions (e.g., competition and predation).
Implications:
Estimation of global extinction risk. Species conservation. Sustainable harvesting strategies.
Moran (1953) analyzed a linear model of two closed populations, subjected to
environmental stochasticity, with no dispersal.
He found correlations between populations equals environmental correlation,
𝜍 𝑧 = 𝜍𝑓 𝑧
Thus, population synchrony scale equals the environmental correlation
length, 𝑚 = 𝑚𝑓
Observations: cohabitant related species with different dispersal abilities
show different synchrony scales. Local dynamics 𝑒𝑂 𝑒𝑢 = 𝑠 𝑂 𝐿 − 𝑂 𝐿 Linear evolution of population fluctuations around equilibrium, 𝜗 = 𝑂 − 𝑂 𝑓𝑟 /𝑂 𝑓𝑟 , with 𝑂 𝑓𝑟 = 𝐿 𝑒𝜗 𝑨, 𝑢 = − 𝑠 + 𝑛 𝜗 𝑨, 𝑢 𝑒𝑢 + 𝑛 𝑒𝑢 𝜗 𝑨 − 𝑦, 𝑢 𝑔 𝑦 𝑒𝑦 + 𝜏𝑓 𝑒𝐶 𝑨, 𝑢 Spatial population synchrony scale (Lande, Engen, and Sæther, 1999) 𝑚2 = 𝑚𝑓
2 + 𝑛 𝑚𝑛 2
𝑠
Evolution equations 𝑒𝑂1 𝑒𝑢 = 𝑠
1𝑂1
𝐿
1 − 𝑂1 − 𝛽1𝑂2
𝐿
1
− 𝛾1 𝑂1 𝑒𝑂2 𝑒𝑢 = 𝑠2𝑂2 𝐿2 − 𝑂2 − 𝛽2𝑂1 𝐿2 − 𝛾2 𝑂2
Stable coexistence when 1/ 𝛽2
∗ 1 − 𝛾1 ∗
> 1 > 𝛽1
∗ 1 − 𝛾2 ∗ , (with 𝛽𝑗 ∗ ≡ 𝛽𝑗 𝐿 𝑘/𝐿𝑗
and 𝛾𝑗
∗ ≡ 𝛾𝑗/𝑠𝑗), with equilibrium point
𝑂1
𝑓𝑟 = 𝐿 1
1 − 𝛾1
∗ − 𝛽1 ∗ 1 − 𝛾2 ∗
1 − 𝛽1
∗ 𝛽2 ∗
𝑂2
𝑓𝑟 = 𝐿2
1 − 𝛾2
∗ − 𝛽2 ∗ 1 − 𝛾1 ∗
1 − 𝛽1
∗ 𝛽2 ∗
Harvesting and competition displace the deterministic equilibrium, reducing the effective carrying capacities.
Procedures for one species models can be generalized. We assume 𝜗𝑗 ≪ 1 (small population fluctuations), 𝛽𝑗
∗ ≪ 1 (inter-specific
competition weaker than intra-specific competition: competitive exclusion principle), and 𝛾𝑗
∗ ≪ 1 (harvesting rates smaller than growth rates).
𝑚1
2 = 𝑚𝑓1 2 + 𝑛1 𝑚𝑛1 2
𝑠
1
1 + 𝛽1
∗ Φ1 + 𝛾1 ∗
+ 𝒫 𝛽1
∗2, 𝛾 1 ∗2, 𝛽1 ∗𝛾 1 ∗
𝑚2
2 = 𝑚𝑓2 2 + 𝑛2 𝑚𝑛2 2
𝑠2 1 + 𝛽2
∗ Φ2 + 𝛾2 ∗
+ 𝒫 𝛽2
∗2, 𝛾2 ∗2, 𝛽2 ∗𝛾2 ∗
(Φ𝑗: competition sensitivity.)
Population synchrony scales
𝑚1
2 = 𝑚𝑓1 2 + 𝑛1 𝑚𝑛1 2
𝑠
1
1 + 𝛾1
∗
+ 𝒫 𝛾
1 ∗2
𝑚2
2 = 𝑚𝑓2 2 + 𝑛2 𝑚𝑛2 2
𝑠2 1 + 𝛾2
∗
+ 𝒫 𝛾2
∗2
Small dispersal contribution ⟹ harvesting does not affect the spatial
synchrony scales.
Large dispersal contribution ⟹ harvesting increases synchrony scale of
both species.
Population synchrony scales
𝑚1
2 = 𝑚𝑓1 2 + 𝑛1 𝑚𝑛1 2
𝑠
1
1 + 𝛽1
∗ Φ1
+ 𝒫 𝛽1
∗2
𝑚2
2 = 𝑚𝑓2 2 + 𝑛2 𝑚𝑛2 2
𝑠2 1 + 𝛽2
∗ Φ2
+ 𝒫 𝛽2
∗2
Small dispersal contribution ⟹ competition does not affect the synchrony
scale.
Large dispersal contribution ⟹ effect of competition depends on sign of the
competition sensitivity.
We distinguish two cases: uncorrelated or correlated environmental noises.
Uncorrelated environmental noises: 𝜍12 𝑧 = 0. Competition sensitivities are Φ1 = Φ2 = 1, i.e.,
𝑚𝑗
2 = 𝑚𝑓𝑗 2 + 𝑛𝑗 𝑚𝑛𝑗 2
𝑠𝑗 1 + 𝛽𝑗
∗ + 𝒫 𝛽𝑗 ∗2
Competition between species with uncorrelated environmental noises always
increases the synchrony scale of both species.
Correlated environmental noises: 𝜍12 𝑧 ≠ 0. A richer situation: Φ1 and Φ2 may be positive or negative. For completely correlated environmental noises (𝜍12 𝑧 = 𝜍1 𝑧 = 𝜍2 𝑧 ),
competition increases the spatial scale of the species with
the larger environmental variance 𝜏𝑓, the larger dispersal capacity 𝑛 𝑚𝑛
2 ,
and the smaller growth rate 𝑠,
while it decreases the spatial scale of the other one. However, when these effects compete the result is not straightforward.
For 𝜏𝑓1 = 𝜏𝑓2 or 𝑠
1 = 𝑠2, never both scales are simultaneously
increased (opposite to uncorrelated noises) For 𝑛1𝑚𝑛1
2
= 𝑛2𝑚𝑛2
2 , for some sets of values
competition increases both synchrony scales (as in the uncorrelated scenarios)
Synchrony scales as function of the ratio
with 𝑠
1 = 𝑠2 = 1, 𝑛1𝑚𝑛1 2
= 𝑛2𝑚𝑛2
2
= 1, 𝛽1
∗ = 𝛽2 ∗ = 0.2 (for competition cases),
and 𝛾1
∗ = 𝛾2 ∗ = 0.2 (for harvesting cases).
Synchrony scales as function of the migration capacity, for species with 𝑠
1 =
2, 𝑠2 = 1, 𝜏𝑓1 = 𝜏𝑓2 = 1, 𝛽1
∗ = 𝛽2 ∗ = 0.2
(for competition cases), and 𝛾1
∗ = 𝛾2 ∗ =
0.2 (for harvesting cases).
Harvesting and inter-species competition affect the spatial synchrony scales of species when the dispersal contribution is relevant. In this case:
Harvesting increases the synchrony scales of both species.
The effect of competition is different for species with uncorrelated environmental noises and for species with correlated environmental noises.
For competing species with uncorrelated environmental noises, competition increases the spatial population synchrony scale of both species.
For competing species with correlated environmental noises, competition increases the synchrony scale of the species with larger environmental variance, larger migration capacity, and smaller growth rate, while decreases the synchrony scale of the other. However, when these effects compete the result is not straightforward, and competition might increase or decrease the synchrony scales of both species.
These effects are relevant for the sustainable exploitation of natural resources and are useful for the study of global extinction risk.