$R_0$ Analysis of a Spatiotemporal Model for a Stream Population

Abstract

Water resources worldwide require management to meet industrial, agricultural, and urban consumption needs. Management actions change the natural flow regime, which impacts the river ecosystem. Water managers are tasked with meeting water needs while mitigating ecosystem impacts. We develop process-oriented advection-diffusion-reaction equations that couple hydraulic flow to population growth, and we analyze them to assess the effect of water flow on population persistence. We present a new mathematical framework, based on the net reproductive rate R0 for advection-diffusion-reaction equations and on related measures. We apply the measures to population persistence in rivers under various flow regimes. This work lays the groundwork for connecting R0 to more complex models of spatially structured and interacting populations, as well as more detailed habitat and hydrological data.

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“…We call R 0 the net reproductive rate, which represents the average number of offspring an individual may produce during its lifetime. Similar to the next generation operator in [25] (see equation (2.13) therein), we can show that the operator defined by (3.10) is a bounded, compact, linear operator on X (see Proposition 2.6 in [25]). Then the Krein-Rutman theorem implies that R 0 is the dominant eigenvalue of the next generation operator Γ with a positive eigenfunction.…”

“…Since the next generation operator Γ has a formula similar to that of the next generation operator defined in [25] (see equation (2.13) therein), applying the results of R 0 analysis in [25] to this paper, we see that if R 0 > 1 (R 0 < 1), then the number Downloaded 05/10/18 to 34.218.44.141. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php of adults in the next generation will be more (less) than the current adults.…”

“…In particular, we allow for two types of boundary conditions relevant to rivers, which are referred to as hostile and Danckwerts' boundary conditions in [25]. Hostile conditions represent zero-flux at the river source (no individuals leave or enter the domain at the upstream boundary) and zero-density at the river outflow (all individuals die at the downstream boundary):…”

“…The solution to (3.8) is a Green's function (see Chapter 7 in [7] and Chapter 3 in [33]). In particular, when q, D, m, σ, and h are constant, by arguments similar to those in [25], we are able to obtain an explicit expression for the Green's function k(x, y) (Appendix B).…”

“…We apply one of the principal projection methods, the collocated method, reviewed in [4, sect. 3.1.1] and restated in [25], to numerically approximate R 0 . The details about this numerical method are provided in section S2 in the supplemental material (M105782SupMat.pdf [local/web 265KB]).…”