Persistence Probabilities for Stream Populations

Abstract

Individuals in streams and rivers are constantly at risk of being washed downstream and thereby lost to their population. The possibility of diffusion-mediated persistence of populations in advective environments has been the focus of a multitude of recent modeling efforts. Most of these recent models are deterministic, and they predict the existence of a critical advection velocity, above which a population cannot persist. In this work, we present a stochastic approach to the persistence problem in streams and rivers. We use the dominant eigenvalue of the advection–diffusion operator to transition from a spatially explicit description to a spatially implicit birth–death process, in which individual washout from the domain appears as an additional death term. We find that the deterministic persistence threshold is replaced by a smooth transition from almost sure persistence to extinction as advection velocity increases. More interestingly, we explore how temporal variation in flow rate and other parameters affect the persistence probability. In line with general expectations, we find that temporal variation often decreases the persistence probability, and we focus on a few examples of how variation can increase population persistence.

Appendix

1.1 A.1 Formulas for the Drift Model

We outline the calculations that lead to the formulae presented in Sect. 4.1. We consider the probability generating function,

$$F(t,x)=\sum_{n=0}^\infty p_n(t)x^n.$$

(29)

Using the set of (5), one can derive the first-order partial differential equation

$$\frac{\partial F}{\partial t} +(\beta x-\mu-\varepsilon ) (1-x)\frac{\partial F}{\partial x}=0,$$

(30)

together with the initial condition $F(0,x)=x^{n_{0}}$. Then we employ the method of characteristics (John 1971) to find the solution F(t,x). The solution F(t,x) is constant along the characteristic curves t(s)=s and

$$\frac{dx}{ds}=(\beta x-\mu-\varepsilon ) (1-x).$$

(31)

This Riccati equation can be solved explicitly. For given t>0, we then find x(0) so that x(t)=0. Then $p_{0}(t)=F(t,0)=F(0,x(0))=x(0)^{n_{0}}$ is the extinction probability as in (18).

To obtain the formula for periodic coefficient functions, we note the following relations. Since

$$\int_0^{kT}\beta(z) \,dz = k \int _0^{T}\beta(z) \,dz,$$

(32)

and similarly for μ(z) and ε(z), we see that N(kT)=N(T)^k. Then we can evaluate

$$\int_0^{kT}\frac{\beta(z)\,dz}{N(z)}=\sum _l^{k-1}\int_0^{T}\frac{\beta(z)\,dz}{N(z+lz)}=\int_0^{T}\frac{\beta(z)\,dz}{N(z)}\sum_l^{k-1}\frac{1}{N(T)^l}.$$

(33)

Together with the formula for the geometric series, this leads to the desired formula (21).

1.2 A.2 Formulas for the Drift-Benthos Model

Following Maler and Lutscher (2010), we present a brief derivation of (7). We denote p _n,m (t) as the probability that there are n drift individuals and m benthic individuals at time t. There are the following seven possibilities for having n drift and m benthic cells at time t+Δt.

• 1. There were n+1 drift and m benthic individuals at time t and one drift individual emigrated. The probability of this event is p _n+1,m (t)(n+1)εΔt+o(Δt).

• 2. There were n drift and m+1 benthic individuals at time t and one benthic individual died: probability p _n,m+1(t)(m+1)μΔt+o(Δt).

• 3. There were n drift and m−1 benthic individuals at time t and one benthic individual gave birth: probability p _n,m−1(t)(n+1)βΔt+o(Δt).

• 4. There were n−1 drift and m+1 benthic individuals at time t and one benthic individual moved to the drift: probability p _n−1,m+1(t)(m+1)τΔt+o(Δt).

• 5. There were n+1 drift and m−1 benthic individuals at time t and one drift individual settled on the benthos: probability p _n+1,m−1(t)(m−1)σΔt+o(Δt).

• 6. There were n drift and m benthic individuals at time t and no events occurred: probability p _n,m (t)(1−n(ε+σ)Δt−m(β+μ+τ)Δt)+o(Δt).

• 7. There were some number of individuals at time t and more than one event occurred: probability o(Δt).

Therefore, p _n,m (t+Δt) is equal to the sum of the seven probabilities given above. Subtracting p _n,m (t) from this expression, dividing by Δt, and letting Δt→0, we get (7).

Now we define P _n =∑ _m p _n,m and Q _m =∑ _n p _n,m , the probability that there be n drift individuals or m benthic individuals, respectively. Summing (7) with respect to m, we obtain the following equation for P _n ,

(34)

Now we form expectations N _d =∑ _n P _n and N _b =∑ _m Q _m . From (34), we obtain an equation for N _d . (Note that ∑ _n ∑ _m p _n,m =1.)

The only term that needs closer consideration is the following:

Hence, we get the equation

$$\dot{N}_d=-\varepsilon N_d-\sigma N_d+\tau N_b.$$

(35)

The derivation of the equation for N _b is similar, so that we arrive at system (9).