StaPOPd: Applied Stable Population Theory for Wildlife Species Reintroduction

The population matrix model

The population matrix model is a stage-structured demographic model that links vital rates in a life history to the population-scale dynamics that they generate (Tuljapurkar, 2008). Also know as the Leslie matrix (age structured; Leslie, 1945) or Lefkovitch (stage structured; Lefkovitch 1965), the population matrix model has been used to enhance understanding and aid management of nearly 8,000 floral and faunal systems across the globe (see Salguero-Gomez et al. 2014, Salguero-Gomez, et al. 2016).

The deterministic population matrix model describes two types of system-level dynamics; ‘asymptotic’ (stable) growth, and ‘transient’ (non-stable) growth. This software suggests reintroduction strategies that are immediately aligned with stable population dynamics with calculations designed to allow populations to theoretically arrive at the predetermined management goals in the desired time frame.  Stage abundances that are out of the stable stage distribution (SSD) will produce non-stable, non-patterned transient dynamics (Fox & Gurevitch 2000) that may help or hurt a reintroduction strategy. Transient population oscillations that cause low abundances may alone increase the likelihood of reintroduction failure.  Reintroduction of stage abundances in concordance with the SSD ensure that the anthropogenic contribution to the population is aligned with a monotonic population trajectory. 

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User Tutorial

Step 1:  Enter all the elements of the matrix model for your system of interest.  Save your model element entries(by clicking on the save button at the bottom). 

Step 2: Click on the "Desired management goals" tab.  Enter your goals, and then click the "Calculate candidate models" button. 

Step 3: Click on the "Candidate models" tab. Row 1 will be your model of interest, while the other rows are pre-saved models with similar life history structure and vital rates.  Enter the row number corresponding to the model that you wish to investigate.  Click on "Calculate detailed information for this model" button.  This click will activate all the remaining tabs.  

Step 4:  View the output on any of the other tabs.  

Step 5: Reintroduction out of SSD tab.  What if I don't have the proportions of individuals that stable population theory recommends?  Enter the abundances that you DO have into the stage-specific boxes.  Once all the boxes are filled with an integer value, the deterministic trajectory from those initial conditions will automatically appear on the plot.  Maybe the abundances at hand will achieve your goals?  Maybe the abundances at hand won't.  The app doesn't make any decisions, it just facilitates investigation from the limited perspective of stable population theory.  

Want to select a different model to investigate?  Return to Step 3.  

Want to select a different life history completely, or modify your original model?  Return to Step 1.  

User Inputs

Model elements: These are the entries of your population matrix model, as parameterized by deterministic, population-scale elements.  The elements in the top row are the fertilities, while the elements in the other rows are the transitions. If a fertility or transition does not exist, enter "0".  Entries must reflect the formal parameterization of a deterministic, non-density dependent, one-sex stage structured model (e.g. see Caswell, 2001).  

Desired unit of time, T: This is an integer above 1, but not more than ~15 (due to technical detail on the math side). The units are consistent with how time in your population matrix model of interest is parameterized.  For example, the time unit could be one year, one month, one day, or one hour.  This unit is up to you.  But the app only takes integers, so please don't type the units!  

Desired female abundances at time T: This is an integer greater than 0 that constitutes the target number of breeding females by the end of the management timeline.  Type an integer into the box.  There is no upper limit.     

Stage abundances for viewing trajectories out of SSD: Enter the number of female individuals in each stage that you would like to release.  Types integers into each box.  These integers must be greater than or equal to 0.   

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This app identifies reintroduction considerations through the limited perspective of demographic stable population theory.  

In particular, the software omits:

• habitat considerations
• genetic considerations
• disease considerations
• density dependence considerations
• sex ratio considerations
• behavior considerations
• all types of variation (demographic, environmental, sampling)
• weather or climate considerations
• species competition considerations
• migration considerations
• predator considerations

• Release abundances hinge on your timeframe and your desired breeding population. Does the app call for too many to be released?  A relaxed timeline or relaxed abundance expectation might be a solution.  
• Do problematic discrepancies exist between goal abundances (of all stages) and the carrying capacity of the intended recipient site?  Can the site support all the animals in all the stages?   
• Do the suggested release abundances constitute a financially realistic captive breeding or live trapping scenario? 
• Play around with all the various inputs.  Try different combinations of approaches in the context of your species of interest.  
• You can scale up or scale down your reintroduction endeavor using stable proportions, and this scaling will indeed maintain stable-status.    

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Please cite this software using:

Hanley, B., Bunting, E., & Schuler, K. 2019. StaPOPd Web Interactive: Interactive software code to "How do we augment the few that remain? Using stable population dynamics to aid reintroduction planning of an iteroparous species" [Software]. Cornell University Library eCommons Repository. https://doi.org/10.7298/d9hg-wa84

Please cite the refereed manuscript using:

Hanley BJ, Bunting EM, Schuler KL. 2019. How can we augment the few that remain? Using stable population dynamics to aid reintroduction planning of an iteroparous species. PeerJ 7:e6873 https://doi.org/10.7717/peerj.6873

The derivations for this app appear in the supplement of the above named work.  The mathematical relationship arises from a fundamental result in population demography using population matrix models (Caswell, 2001).  All we did was symbolically solve a recursion so these foundational equations could explicitly intake situational management goals, and then make the solution more accessible.  

Foundational citations

Caswell, H. (2001). Matrix population models:  Construction, analysis, and interpretation.  2nd Ed.  Sinauer, Sunderland, Massachusetts, USA.  ISBN: 9780878931217

To see just how important population matrix models are to ecology and management, visit:

Salguero-Gómez R, Jones OR, Archer CR, Bein C, de Buhr H, Farack C, Gottschalk F, Hartmann A, Henning A, Hoppe G, Römer G, Ruoff T, Sommer V, Wille J, Voigt J, Zeh S, Vieregg D, Buckley YM, Che-Castaldo J, Hodgson D, Scheuerlein A, Caswell H, Vaupel JW, & Coulson T (2016). COMADRE: a global data base of animal demography, Journal of Animal Ecology, 85(2), 371. DOI: 10.1111/1365-2656.12482

and

Salguero-Gómez R, Jones OR, Archer CR, Buckley YM, Che-Castaldo J, Caswell H, Hodgson D, Scheuerlein A, Conde DA, Brinks E, de Buhr H, Farack C, Gottschalk F, Hartmann A, Henning A, Hoppe G, Romer G, Runge J, Ruoff T, Wille J, Zeh S, Davison R, Vieregg D, Baudisch A, Altwegg R, Colchero F, Dong M, de Kroon H, Lebreton JD, Metcalf CJE, Neel MM, Parker IM, Takada T, Valverde T, Velez-Espino LA, Wardle GM, Franco M, Vaupel JW (2014). The COMPADRE plant matrix database: an open online repository for plant demography.  Journal of Ecology, 103(1), pgs 202-218.  DOI: 10.1111/1365-2745.12334

and for a helpful individual-based PVA reintroduction model, see:

Vortex: Lacy, R.C., and J.P. Pollak. 2014. Vortex: A stochastic simulation of the extinction process. Version 10.0. Chicago Zoological Society, Brookfield, Illinois, USA.

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