Understand the relationship
The reasoning behind the result
A matrix connects current sources to next-census destinations
n_i(t+1) = Σ_j A_ij n_j(t)
The vector n contains one population quantity for each entered stage or age class. A coefficient Aij describes the next-census contribution to destination i per individual currently in source j. Each destination adds contributions across its row; a source's contributions appear down its column.
The stage names and ordering must match the initial vector and every matrix. A Leslie age-class model is one structured form of this relationship; a general stage model can also include remaining in the same stage or moving between other classes. The calculator does not infer which transitions are biologically possible.
Retained individuals and new recruits have different meanings
A = U + F
U describes the fate of individuals already present. A column sum below one means some source individuals are not retained in the represented population at the next census. A sum above one would count an existing individual in more than one mutually exclusive destination, so separate-U mode rejects it apart from tiny addition roundoff.
F describes new recruits present at the next census per current source individual. Recruitment can exceed one, so F and the combined A are not probability matrices. A combined matrix alone cannot uniquely distinguish retained individuals from recruits; the output only makes that separation when the visitor supplies U and F.
Census timing belongs inside the coefficients
A birth count immediately after reproduction and a recruit count surviving to a later census are different quantities. If the intended projection counts individuals at the later census, the supplied F must already include survival to that point. The calculator does not silently multiply by another survival factor or assume a sex ratio.
State the census timing and duration before entering coefficients. A matrix defined per year cannot be applied every month simply by changing the time label, and a multi-year interval is not automatically one annual transition. Each iteration applies the complete supplied matrix exactly once.
Finite composition and total growth need not behave alike
Total(t) = Σ_i n_i(t); stage share = 100 n_i/Total
Different initial compositions can produce different near-term total changes under the same matrix. A total multiplier between two consecutive censuses is a result of that particular vector and transition, not automatically a stable long-run growth factor.
The chart and tables show each complete census within the requested horizon. The first and final step ledgers expose every source-to-destination contribution. Connecting chart segments guide the eye; the matrix does not specify continuous movement within an interval. A zero total leaves stage percentages undefined.
Unrepresented processes stay outside the model
The projection holds coefficients fixed and includes only the stage relationships entered in A or U+F. It does not introduce density dependence, immigration, environmental variation or uncertainty. Individuals not retained by U are labelled as such because their absence can have more than one cause.
Fractional outputs are expected quantities under the supplied arithmetic, not promises about whole individuals. The calculator does not infer an equilibrium, stable-stage distribution, extinction risk or population viability from a finite trajectory. Those conclusions require additional model assumptions and evidence.
Follow the numbers
Each destination adds its source contributions
- Start with stage vector [100,200] and rows [0.8,1.2] and [0.6,0.5].
- At the next census, stage A receives 0.8×100+1.2×200=320. Stage B receives 0.6×100+0.5×200=160.
- Apply the same matrix again: stage A=0.8×320+1.2×160=448; stage B=0.6×320+0.5×160=272. The second-census total is 720.
The off-diagonal coefficients contribute from a different source stage. Reversing rows and columns would produce a different projection.
Quick guide
How to use this calculator
- Choose a combined matrix or separate retention and recruitment matrices, and set the stage count.
- State the census timing, interval duration and complete number of periods.
- Enter initial stage quantities and matrices in the same order: columns are sources and rows are destinations.
- Inspect each census, the first and last step's contributions, and the final composition. Treat the trajectory as an entered model rather than a biological prediction.
Calculation method
Calculation and interpretation
Show how source-stage individuals contribute to each next-census stage.
n(t+1)=A n(t); A=U+F; destination i receives Σj Aij nj from all source stages j
Worked example
Each destination adds its source contributions
The off-diagonal coefficients contribute from a different source stage. Reversing rows and columns would produce a different projection.
n(t+1)=A n(t); A=U+F; destination i receives Σj Aij nj from all source stages j
Supported inputs
Precision and limits
Entered parameters and census convention
The model supplies no species-specific survival, fecundity, sex ratio, migration or biological recommendations. Recruitment coefficients must already match the stated census timing.
Finite deterministic scenario
No carrying capacity, stable-stage distribution, long-run growth guarantee, extinction risk or population viability is inferred. Constant coefficients can be inappropriate when conditions change.
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